Choose 2D or 3D, then enter the components of the two vectors a and b and the coefficients k and l. The formula below is linked to the input fields, so you can also edit the components right inside it.
Enter numbers only. Decimals, negative numbers and fractions such as 3/4 are OK. A blank component is treated as 0, and a blank coefficient k or l as 1.
Result and figure
Enter the components of the vectors in the fields on the left and press "Calculate". The result and a figure will appear here.
Table of Contents
-
What you can do on this page
-
What is this calculation used for?
-
How to Use
-
Formulas and figures
-
Magnitude of a vector (from its components)
-
Sum and difference of vectors (add or subtract component by component)
-
Scalar multiple of a vector
-
Dot product (a calculation that turns two vectors into a number)
-
Angle between two vectors
-
Conditions for perpendicular and parallel vectors
-
Unit vector (keeping only the direction)
-
-
Symbols and terms
-
Good to know before you start
-
How to calculate it in Excel
-
How to calculate it in Google Sheets
-
How to calculate it in Python
-
How to write it in LaTeX and other math languages (copy and paste)
-
How to have ChatGPT do the calculation
-
DataChef Features
-
Related Features
-
NumberChef Calculators List
What you can do on this page
- Just enter the components of the 2D or 3D vectors \(\vec{a}\) and \(\vec{b}\) to get the sum \(\vec{a}+\vec{b}\), the difference \(\vec{a}-\vec{b}\) and the linear combination \(k\vec{a}+l\vec{b}\) at once
- The magnitude \(|\vec{a}|\) is shown both as an exact value with the radical simplified, such as \(\sqrt{20}=2\sqrt{5}\), and as a decimal
- You also get the dot product \(\vec{a}\cdot\vec{b}\), the angle between the vectors \(\theta\) (in degrees and radians, with exact values for special angles such as \(45^{\circ}\)), the unit vectors and a perpendicular/parallel check
- The calculator shows the vectors as columns, just as you would write them by hand, and you can edit the components right inside them
- Components and coefficients can be decimals, negative numbers or fractions such as 3/4
- A plain-language explanation of the formulas and copy-and-paste formulas for Excel, Google Sheets and Python are all on this page
A vector with magnitude 0 (the zero vector) has no direction, so there is no angle between it and another vector and no unit vector. This calculator does not calculate when either \(\vec{a}\) or \(\vec{b}\) is the zero vector.
What is this calculation used for?
Combining forces (physics, architecture and engineering)
When several forces act on an object, their combined effect is the sum of vectors. Splitting the tension of a rope pulled at an angle into horizontal and vertical components and adding them up is exactly this calculation.
In designing bridges and buildings, forces such as the structure's own weight, wind and earthquakes are combined as vectors to find the direction and size of the force on each part. Adding only the sizes and ignoring the directions would lead to a dangerous design, so working with vectors is essential.
3D games and computer graphics
In a game, a character's movement is calculated as "position vector + velocity vector", and the direction of the camera and of a flying projectile are all vectors. Unit vectors are used every frame whenever only a direction is needed (direction of travel, direction of view).
The dot product is also the key to lighting. The larger the dot product of the direction a surface faces (its normal vector) and the direction of the light, the more directly the light hits it, and that value sets the brightness on screen. If the dot product is 0 or less, the light does not reach the surface.
The course of a plane or ship (combining velocities)
A plane's actual path is the sum of its velocity vector in the direction the nose points and the wind's velocity vector. To fly due east with a wind from the north, the nose has to point partly into the wind, and how far to turn it is found by subtracting vectors.
At sea, ships add and subtract the vector of the current to decide which way to steer. Being able to handle speed and time component by component is a strength of vectors.
Measuring how similar texts or products are (AI and recommendations)
Search engines and generative AI turn texts into long lists of numbers, that is, vectors. Whether two texts are similar is judged by the angle between their vectors: the closer \(\cos\theta\) is to 1, the more similar they are (cosine similarity).
Online stores' "customers who viewed this also liked" features also turn purchase histories into vectors and look for people and products with small angles between them. Here the direction (the pattern of tastes) matters more than the length (such as how much someone spends in total), which is why measuring by angle works well.
Devices that measure position and tilt (drones, GPS navigation and smartphones)
Drones and smartphones work out how much they are tilted from the angle between the gravity vector measured by the accelerometer and a reference direction vector of the device. The screen switching between portrait and landscape is a result of this calculation.
In GPS navigation and surveying, the difference between the coordinates of two points is treated as a vector: its magnitude is the distance and its direction is the bearing. In 3D, just add the height component and the same formulas work as is.
Formulas and figures
Magnitude of a vector (from its components)
Figure
Standard notation
(the usual math form)
\(|\vec{a}|\)
\(=\)
\(\sqrt{a_1^{2}+a_2^{2}}\)
In words
(symbols replaced with words)
②
\(|\vec{a}|\): magnitude of vector \(\vec{a}\) (length of the arrow)
\(=\)
①
square root of the sum of the squared components, \(a_1^{2}+a_2^{2}\)
The formula in words
①
When \(\vec{a}=(a_1,\ a_2)\) in component form, take the
square root of \(a_1^{2}+a_2^{2}\), the \(x\) component squared plus the \(y\) component squared
,
②
and you get the
\(|\vec{a}|\): magnitude of vector \(\vec{a}\) (length of the arrow)
Quick example
The magnitude of \(\vec{a}=(4,\ 2)\) (the length of an arrow that goes 4 right and 2 up) is
magnitude of \(\vec{a}=(4,\ 2)\)
\(=\)
square root of \(4^2+2^2=20\)
\(|\vec{a}| = \sqrt{4^{2}+2^{2}} = \sqrt{20} = 2\sqrt{5} \approx 4.4721\)
Key idea
This formula is the Pythagorean theorem itself. The components \(a_1\) and \(a_2\) split the arrow into "how far right" and "how far up", so the arrow is the hypotenuse of a right triangle whose two legs are \(a_1\) and \(a_2\). That is why its length is \(\sqrt{a_1^{2}+a_2^{2}}\).
For a 3D vector \(\vec{a}=(a_1,\ a_2,\ a_3)\), add the square of the height component too: \(|\vec{a}|=\sqrt{a_1^{2}+a_2^{2}+a_3^{2}}\). There is just one more term; the steps are the same.
Always write the radical in the answer in simplest form (for example, \(\sqrt{20}=2\sqrt{5}\)). This calculator also shows both the simplified exact value and a decimal.
Sum and difference of vectors (add or subtract component by component)
Figure
Standard notation
(the usual math form)
\(\vec{a}+\vec{b}\)
\(=\)
\((a_1+b_1,\ a_2+b_2)\)
\(\vec{a}-\vec{b}\)
\(=\)
\((a_1-b_1,\ a_2-b_2)\)
In words
(symbols replaced with words)
②
sum of \(\vec{a}\) and \(\vec{b}\)
\(=\)
①
vector made by adding the \(x\) components and the \(y\) components
④
difference of \(\vec{a}\) and \(\vec{b}\)
\(=\)
③
vector made by subtracting the \(x\) components and the \(y\) components
The formula in words
①
The
vector made by adding the \(x\) components and the \(y\) components, \((a_1+b_1,\ a_2+b_2)\)
is the
②
sum of \(\vec{a}\) and \(\vec{b}\), \(\vec{a}+\vec{b}\)
, and the
③
vector made by subtracting the \(x\) components and the \(y\) components, \((a_1-b_1,\ a_2-b_2)\)
is the
④
difference of \(\vec{a}\) and \(\vec{b}\), \(\vec{a}-\vec{b}\)
Quick example
When \(\vec{a}=(4,\ 2)\) and \(\vec{b}=(-1,\ 2)\)
\(\vec{a}+\vec{b}\)
\(=\)
\((4+(-1),\ 2+2)\)
\(\vec{a}+\vec{b} = (4+(-1),\ 2+2) = (3,\ 4)\)
\(\vec{a}-\vec{b} = (4-(-1),\ 2-2) = (5,\ 0)\)
Key idea
In the figure, the sum \(\vec{a}+\vec{b}\) is the arrow you get by placing the start of \(\vec{b}\) at the end of \(\vec{a}\), going from the very first start to the very last end. It is the diagonal of the parallelogram with sides \(\vec{a}\) and \(\vec{b}\), which is why this is called the parallelogram law.
The difference \(\vec{a}-\vec{b}\) is \(\vec{a}+(-\vec{b})\): reverse \(\vec{b}\), then add. In the figure, it is the arrow from the end of \(\vec{b}\) to the end of \(\vec{a}\). The direction is easy to get wrong, so remember: "from the end of the vector being subtracted (\(\vec{b}\)) to the end of the one it is subtracted from (\(\vec{a}\))". If you keep in mind that the arrow points toward \(\vec{a}\), you will not draw it backward.
In 3D, there is just one more term that adds or subtracts the \(z\) components; the steps are the same.
Scalar multiple of a vector
Figure
Standard notation
(the usual math form)
In words
(symbols replaced with words)
\(k\vec{a}\)
\(=\)
\((ka_1,\ ka_2)\)
②
\(k\) times \(\vec{a}\)
\(=\)
①
vector with each component multiplied by \(k\)
The formula in words
①
The
vector with each component multiplied by \(k\), \((ka_1,\ ka_2)\)
is
②
\(k\) times the vector \(\vec{a}\), \(k\vec{a}\)
Quick example
Doubling \(\vec{a}=(4,\ 2)\) (an arrow in the same direction, twice as long) gives
\(2\vec{a}\)
\(=\)
\((2 \times 4,\ 2 \times 2)\)
\(2\vec{a} = (2 \times 4,\ 2 \times 2) = (8,\ 4)\)
\(|2\vec{a}| = \sqrt{8^{2}+4^{2}} = \sqrt{80} = 4\sqrt{5} = 2 \times 2\sqrt{5} = 2|\vec{a}|\)
Key idea
A scalar multiple stretches or shrinks the arrow. If \(k>0\), the direction stays the same and the length becomes \(k\) times as long. If \(k<0\), the direction flips and the length becomes \(|k|\) times as long (\(k=-1\) gives the opposite vector \(-\vec{a}\)). The magnitude is \(|k\vec{a}|=|k|\,|\vec{a}|\).
This calculator works in the form \(k\vec{a}+l\vec{b}\). With \(k=1,\ l=1\) you get the sum, and with \(k=1,\ l=-1\) the difference, so just by changing the coefficients you can also calculate expressions such as \(2\vec{a}-3\vec{b}\).
With \(k=0\), you get the zero vector \(\vec{0}\), whose components are all 0. The zero vector has length 0 and no direction.
Dot product (a calculation that turns two vectors into a number)
Figure
Standard notation
(the usual math form)
\(\vec{a}\cdot\vec{b}\)
\(=\)
\(|\vec{a}|\)
\(\times\)
\(|\vec{b}|\)
\(\times\)
\(\cos\theta\)
\(\vec{a}\cdot\vec{b}\)
\(=\)
\(a_1b_1+a_2b_2\)
In words
(symbols replaced with words)
④
dot product of \(\vec{a}\) and \(\vec{b}\)
\(=\)
①
magnitude of \(\vec{a}\)
\(\times\)
②
magnitude of \(\vec{b}\)
\(\times\)
③
\(\cos\theta\): cosine of the angle \(\theta\)
⑥
dot product of \(\vec{a}\) and \(\vec{b}\)
\(=\)
⑤
product of the \(x\) components plus product of the \(y\) components
The formula in words
①
Multiply the
magnitude of \(\vec{a}\), \(|\vec{a}|\)
, the
②
magnitude of \(\vec{b}\), \(|\vec{b}|\)
and the
③
\(\cos\theta\): cosine of the angle \(\theta\)
to get the
④
dot product \(\vec{a}\cdot\vec{b}\)
. In components, the
⑤
product of the \(x\) components plus product of the \(y\) components, \(a_1b_1+a_2b_2\)
gives the same value, so it also gives the
⑥
dot product \(\vec{a}\cdot\vec{b}\)
Quick example
The dot product of \(\vec{a}=(4,\ 2)\) and \(\vec{b}=(-1,\ 2)\) is
dot product \(\vec{a}\cdot\vec{b}\)
\(=\)
\(4 \times (-1) + 2 \times 2\)
\(\vec{a}\cdot\vec{b} = 4 \times (-1) + 2 \times 2 = -4 + 4 = 0\)
Key idea
The answer of a dot product is not a vector but just a number (a scalar). The name sounds like multiplication, but think of it as a special calculation that turns two arrows into one number. Always write it with a centered dot; \(\vec{a}\times\vec{b}\) is a different calculation (the cross product).
In the figure, it is "the length of the shadow of \(\vec{b}\) cast straight down onto the direction of \(\vec{a}\) (the projection), \(|\vec{b}|\cos\theta\), times \(|\vec{a}|\)". So the more the two directions line up, the larger the dot product. At a right angle the shadow has length 0, so the dot product is 0. When the directions are nearly opposite, the shadow points the other way, so the dot product is negative.
Use the definition (magnitudes and angle) when you know the angle, and the component formula when you know the coordinates. In 3D, a term \(a_3b_3\) is added: \(\vec{a}\cdot\vec{b}=a_1b_1+a_2b_2+a_3b_3\).
Angle between two vectors
Figure
Standard notation
(the usual math form)
\(\cos\theta\)
\(=\)
\(\dfrac{\vec{a}\cdot\vec{b}}{|\vec{a}|\,|\vec{b}|}\)
In words
(symbols replaced with words)
②
\(\cos\theta\): cosine of the angle \(\theta\)
\(=\)
①
\(\dfrac{\text{dot product }\vec{a}\cdot\vec{b}}{\text{product of the magnitudes }|\vec{a}|\,|\vec{b}|}\)
The formula in words
①
Calculate
\(\dfrac{\text{dot product }\vec{a}\cdot\vec{b}}{\text{product of the magnitudes }|\vec{a}|\,|\vec{b}|}\)
to get the
②
\(\cos\theta\): cosine of the angle \(\theta\)
, and from it the angle \(\theta\)
Quick example
The angle between \(\vec{a}=(1,\ 1)\) and \(\vec{b}=(2,\ 0)\) (the dot product is \(1 \times 2 + 1 \times 0 = 2\), and the magnitudes are \(|\vec{a}|=\sqrt{2}\) and \(|\vec{b}|=2\)) is
\(\cos\theta\)
\(=\)
\(\dfrac{2}{\sqrt{2} \times 2}\)
\(\cos\theta = \dfrac{2}{\sqrt{2} \times 2} = \dfrac{1}{\sqrt{2}} = \dfrac{\sqrt{2}}{2}\)
\(\theta = 45^{\circ} = \dfrac{\pi}{4}\)
Key idea
This is just the definition of the dot product, \(\vec{a}\cdot\vec{b}=|\vec{a}||\vec{b}|\cos\theta\), solved for \(\cos\theta\). Note that what you get first is the value of \(\cos\theta\), not the angle itself.
To get the angle from \(\cos\theta\), compare it with the values for special angles: \(\dfrac{\sqrt{2}}{2}\) is \(45^{\circ}\), \(\dfrac{1}{2}\) is \(60^{\circ}\), and \(0\) is \(90^{\circ}\). For other values, use the inverse cosine \(\cos^{-1}\) (arccos); this calculator shows the result as a decimal. By convention, the angle between two vectors is given in the range \(0^{\circ} \le \theta \le 180^{\circ}\).
The sign of \(\cos\theta\) alone tells you the type of angle: positive is acute, 0 is a right angle, and negative is obtuse. If a radical is left in the denominator, rationalize it, as in \(\dfrac{1}{\sqrt{2}}=\dfrac{\sqrt{2}}{2}\).
Conditions for perpendicular and parallel vectors
Figure
Standard notation
(the usual math form)
\(\vec{a}\perp\vec{b}\)
\(\iff\)
\(\vec{a}\cdot\vec{b}=0\)
\(\vec{a}\parallel\vec{b}\)
\(\iff\)
\(\vec{b}=k\vec{a}\)
In words
(symbols replaced with words)
①
\(\vec{a}\) and \(\vec{b}\) are perpendicular
\(\iff\)
②
the dot product is 0
③
\(\vec{a}\) and \(\vec{b}\) are parallel
\(\iff\)
④
one is a scalar multiple of the other
The formula in words
①
When neither \(\vec{a}\) nor \(\vec{b}\) is the zero vector,
\(\vec{a}\) and \(\vec{b}\) meet at a right angle
exactly when
②
the dot product is 0, \(\vec{a}\cdot\vec{b}=0\)
, and
③
\(\vec{a}\) and \(\vec{b}\) are parallel
exactly when
④
\(\vec{b}=k\vec{a}\) for some real number \(k\)
Quick example
When \(\vec{a}=(4,\ 2)\) and \(\vec{b}=(-1,\ 2)\), the dot product is \(4 \times (-1) + 2 \times 2 = 0\), so
\(\vec{a}\) and \(\vec{b}\) are perpendicular
\(\iff\)
the dot product is \(0\)
\(\vec{a}\cdot\vec{b} = 4 \times (-1) + 2 \times 2 = 0 \quad \Rightarrow \quad \vec{a}\perp\vec{b}\)
\(\vec{c} = (8,\ 4) = 2\vec{a} \quad \Rightarrow \quad \vec{c}\parallel\vec{a}\)
Key idea
To check for perpendicular vectors, you do not need the angle; just check whether the dot product is 0. Since \(\cos 90^{\circ}=0\), the whole definition \(|\vec{a}||\vec{b}|\cos\theta\) becomes 0. It is a standard tool for showing that an angle is a right angle in geometry proofs.
In the plane, the parallel condition \(\vec{b}=k\vec{a}\) written in components is \(a_1b_2-a_2b_1=0\). This is handy because you can check it without finding \(k\) (it says the same thing as the ratios of the components being equal, \(a_1:a_2=b_1:b_2\)).
Both conditions assume that neither \(\vec{a}\) nor \(\vec{b}\) is the zero vector. The zero vector has a dot product of 0 with every vector, but since it has no direction, we do not call it perpendicular, and we do not ask whether it is parallel. That is why textbooks state the conditions with "when \(\vec{a} \neq \vec{0},\ \vec{b} \neq \vec{0}\)".
Unit vector (keeping only the direction)
Figure
Standard notation
(the usual math form)
\(\dfrac{\vec{a}}{|\vec{a}|}\)
\(=\)
\(\left(\dfrac{a_1}{|\vec{a}|},\ \dfrac{a_2}{|\vec{a}|}\right)\)
In words
(symbols replaced with words)
②
vector with the same direction as \(\vec{a}\) and magnitude 1
\(=\)
①
\(\dfrac{\text{original components }(a_1,\ a_2)}{\text{magnitude of the vector }|\vec{a}|}\)
The formula in words
①
Divide each component by the magnitude, as in
\(\dfrac{\text{original components }(a_1,\ a_2)}{\text{magnitude of the vector }|\vec{a}|}\)
, to get the
②
vector with the same direction as \(\vec{a}\) and magnitude 1 (the unit vector)
Quick example
The unit vector in the direction of \(\vec{a}=(4,\ 2)\) (with magnitude \(|\vec{a}|=2\sqrt{5}\)) is
unit vector in the direction of \(\vec{a}=(4,\ 2)\)
\(=\)
\(\dfrac{(4,\ 2)}{2\sqrt{5}}\)
\(\dfrac{\vec{a}}{|\vec{a}|} = \left(\dfrac{4}{2\sqrt{5}},\ \dfrac{2}{2\sqrt{5}}\right) = \left(\dfrac{2\sqrt{5}}{5},\ \dfrac{\sqrt{5}}{5}\right)\)
\(\sqrt{\left(\dfrac{2\sqrt{5}}{5}\right)^{2}+\left(\dfrac{\sqrt{5}}{5}\right)^{2}} = \sqrt{\dfrac{20}{25}+\dfrac{5}{25}} = \sqrt{\dfrac{25}{25}} = 1\)
Key idea
A unit vector is a vector that drops the length information and keeps only the direction. Dividing by the magnitude always gives length 1, because \(\left|\dfrac{\vec{a}}{|\vec{a}|}\right| = \dfrac{|\vec{a}|}{|\vec{a}|} = 1\), so you do not need to check the length again after dividing.
With the direction alone, you can write "the point 5 units along the direction of \(\vec{a}\)" as "5 × the unit vector". It is a tool for cases where length does not matter and only direction does, such as the direction of travel, the direction of view or the direction a surface faces (the normal).
Dividing the components by the magnitude puts a radical in the denominator, so rationalize the answer, as in \(\dfrac{4}{2\sqrt{5}}=\dfrac{2\sqrt{5}}{5}\). There is a vector of magnitude 1 for every direction, so say "in the direction of \(\vec{a}\)" to tell it apart from the unit vector in the opposite direction, \(-\dfrac{\vec{a}}{|\vec{a}|}\). In 3D, there is just one more term that divides the \(z\) component by the same magnitude; the steps are the same.
A vector is a quantity with both direction and magnitude, and writing it in component form, \(\vec{a}=(a_1,\ a_2)\), makes the calculations much easier. Sums, differences and scalar multiples are done component by component, the magnitude is \(|\vec{a}|=\sqrt{a_1^{2}+a_2^{2}}\) (the Pythagorean theorem), and the dot product is the sum of the products of the components, \(a_1b_1+a_2b_2\). Dividing the dot product by the product of the magnitudes gives \(\cos\theta\), which tells you the angle between the two vectors. If the dot product is 0, the vectors are perpendicular; if one is a scalar multiple of the other, they are parallel.
Symbols and terms
Symbols
| \(\vec{a}\) | vector a | The symbol for a vector. The arrow over the letter shows that it is a quantity with direction. Some textbooks use bold type instead, as in \(\boldsymbol{a}\); both are the same vector. The directed segment from point A to point B is written \(\overrightarrow{\mathrm{AB}}\) (the vector from A to B). |
| \(a_1,\ a_2,\ a_3\) | a sub one, a sub two, a sub three | The components of vector \(\vec{a}\). The small number at the lower right (the subscript) tells which component it is: \(a_1\) is the step in the \(x\) direction, \(a_2\) in the \(y\) direction and \(a_3\) in the \(z\) direction. Some textbooks use \(x\) and \(y\) as subscripts instead, as in \(\vec{a}=(a_x,\ a_y)\). |
| \(|\vec{a}|\) | magnitude of vector a | The magnitude of vector \(\vec{a}\) (the length of the arrow). It uses the same vertical bars as the absolute value of a real number, with the same idea: drop the direction and look only at the size. The value is always 0 or more. |
| \(\vec{a}\cdot\vec{b}\) | a dot b | The dot product. By convention it is written with a centered dot, and the answer is a number, not a vector. The name "dot product" comes from this symbol. It is also called the scalar product. |
| \(\theta\) | theta | The angle between two vectors. Greek letters are customary for angles, and \(\theta\) is the most common. The angle between vectors is the opening between them when they start at the same point, taken in the range \(0^{\circ}\) to \(180^{\circ}\). |
| \(\cos\theta\) | cosine theta | The cosine of the angle \(\theta\). In a right triangle, it is the adjacent side divided by the hypotenuse, and it takes values from \(-1\) to \(1\). It is the key link between the dot product and the angle. |
| \(\cos^{-1}\) | inverse cosine (arccosine) | The inverse cosine. A function that gives the angle \(\theta\) from the value of \(\cos\theta\), also written \(\arccos\). The \(-1\) at the upper right stands for "the reverse operation", not \(\cos\) to the power of \(-1\). On a calculator, it is the key marked small above cos (often reached with 2nd or SHIFT). |
| \(k,\ l\) | k, l | Real numbers (scalars) that say how many times to multiply a vector. The letter \(k\) is often used for constants. In this calculator you enter them as the coefficients of \(k\vec{a}+l\vec{b}\). |
| \(\vec{0}\) | zero vector | The vector whose components are all 0 (the zero vector). It is a special vector with magnitude 0 and no direction, so there is no angle or unit vector for it. |
| \(\perp\) | is perpendicular to | The symbol for "is perpendicular to". \(\vec{a}\perp\vec{b}\) is read "\(\vec{a}\) is perpendicular to \(\vec{b}\)". The symbol is a drawing of a right angle. |
| \(\parallel\) | is parallel to | The symbol for "is parallel to". \(\vec{a}\parallel\vec{b}\) is read "\(\vec{a}\) is parallel to \(\vec{b}\)". The symbol is two parallel lines side by side. |
| \(\iff\) | if and only if | A symbol that says the left side and the right side are equivalent. It says "the right side is true when the left side is true, and only then", and it is written as a two-way arrow. |
| \(\sqrt{\ }\) | square root | The square root symbol. \(\sqrt{20}\) is "the positive number that gives 20 when squared". Vector magnitudes are written with this symbol, and answers are given in simplest radical form, pulling square factors out of the radical, as in \(\sqrt{20}=2\sqrt{5}\). |
Terms
| vector | A quantity that has both a direction and a size (magnitude). It describes things such as velocity, force and movement, where the size alone is not enough and you also need the direction. It is drawn as an arrow, and moving it without turning it (same direction and length) gives the same vector. |
| scalar | Just a number with no direction (such as a length, weight or temperature). The word is used to tell it apart from a vector. The answer of a dot product is a scalar. |
| directed line segment | A line segment with a direction: the arrow itself, with a set starting point and end point. A vector is what you get by keeping only its direction and length, whatever its position. |
| component form | Writing a vector as a set of numbers that says "how far in the \(x\) direction, how far in the \(y\) direction". Besides writing it in a row, as in \(\vec{a}=(4,\ 2)\), you can also write it as a column vector; the two are equivalent. Many US textbooks use angle brackets, as in \(\langle 4,\ 2\rangle\). In component form, sums and dot products are just number calculations. |
| magnitude | The length of the vector's arrow. It is also called the norm or length, and you find it from the components with \(\sqrt{a_1^{2}+a_2^{2}}\). |
| unit vector | A vector with magnitude 1. Calculating \(\dfrac{\vec{a}}{|\vec{a}|}\) gives a vector in the same direction as \(\vec{a}\) with its length set to 1. It is used when you only want to show a direction. |
| zero vector | The vector \(\vec{0}\) whose components are all 0. Its magnitude is 0 and it has no direction. It appears as the difference of a vector and itself, as in \(\vec{a}-\vec{a}=\vec{0}\). |
| opposite vector | The vector \(-\vec{a}\) with the same magnitude and exactly the opposite direction. It is the scalar multiple with \(k=-1\), and it is used in the difference \(\vec{a}-\vec{b}=\vec{a}+(-\vec{b})\). |
| scalar multiple | A vector multiplied by a number \(k\) (scalar multiplication). The length becomes \(k\) times as long in the same direction, and if \(k\) is negative the direction reverses. |
| parallelogram law | The rule that the sum of two vectors is the diagonal of the parallelogram with those two vectors as sides. It is used directly to combine forces in a diagram. |
| dot product | A calculation that turns two vectors into one number. It is defined as \(|\vec{a}||\vec{b}|\cos\theta\), and in components it is \(a_1b_1+a_2b_2\). It shows as a number how well the two directions line up. |
| angle between two vectors | The angle \(\theta\) formed when two vectors start at the same point. It is taken in the range \(0^{\circ}\) to \(180^{\circ}\). |
| projection | The shadow of one vector cast onto the direction of another by light shining straight down (the orthogonal projection). The length of the projection of \(\vec{b}\) onto the direction of \(\vec{a}\) is \(|\vec{b}|\cos\theta\) (the scalar projection), and the dot product can be read as this shadow length times \(|\vec{a}|\). |
| perpendicular | Two vectors meeting at a right angle (also called orthogonal). For two vectors that are not the zero vector, it is the same as the dot product being 0. |
| parallel | Two vectors pointing in the same direction or exactly opposite directions. It is the same as one being a scalar multiple of the other, and in the plane you can check it from the components alone with \(a_1b_2-a_2b_1=0\). |
| position vector | A vector that starts at the origin and shows the position of a point. The position vector of point A is \(\overrightarrow{\mathrm{OA}}\), and the coordinates of the point are its components. It is the starting point for studying shapes with vectors. |
Good to know before you start
Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.
If you get stuck, going back to review these topics is the fastest way forward.
| The coordinate plane and 3D coordinates (Grade 6 to Precalculus) |
|
| Arithmetic with negative numbers (Grade 7) |
|
| Square roots (Grade 8 to Algebra 1) |
|
| The Pythagorean theorem (Grade 8) |
|
| Trigonometry - cosine (Geometry) |
|
| Equations and fractions (Grades 6–8) |
|
How to calculate it in Excel
Copy the whole table below and paste it into cell A1 in Excel. It works as is.
Table to find the magnitude of a vector
| x component a1 | 4 |
| y component a2 | 2 |
| z component a3 (0 for 2D) | 4 |
| Magnitude |a| | =SQRT(B1^2+B2^2+B3^2) |
Table to find the sum and difference of vectors
| a, x component a1 | 4 |
| a, y component a2 | 2 |
| a, z component a3 | 4 |
| b, x component b1 | -1 |
| b, y component b2 | 2 |
| b, z component b3 | 2 |
| Sum, x component | =B1+B4 |
| Sum, y component | =B2+B5 |
| Sum, z component | =B3+B6 |
| Difference, x component | =B1-B4 |
| Difference, y component | =B2-B5 |
| Difference, z component | =B3-B6 |
Table to find a scalar multiple of a vector
| Scalar k | 2 |
| x component a1 | 4 |
| y component a2 | 2 |
| z component a3 | 4 |
| k times, x component | =B1*B2 |
| k times, y component | =B1*B3 |
| k times, z component | =B1*B4 |
Table to find the dot product
| a, x component a1 | 4 |
| a, y component a2 | 2 |
| a, z component a3 | 4 |
| b, x component b1 | -1 |
| b, y component b2 | 2 |
| b, z component b3 | 2 |
| Dot product a·b | =B1*B4+B2*B5+B3*B6 |
Table to find the angle between two vectors
| Dot product a·b | 8 |
| Magnitude |a| | 6 |
| Magnitude |b| | 3 |
| cosθ | =B1/(B2*B3) |
| Angle θ (degrees) | =DEGREES(ACOS(B4)) |
| Angle θ (radians) | =ACOS(B4) |
Table to check the conditions for perpendicular and parallel (2D)
| a, x component a1 | 4 |
| a, y component a2 | 2 |
| b, x component b1 | -1 |
| b, y component b2 | 2 |
| Dot product a·b | =B1*B3+B2*B4 |
| Perpendicular? | =IF(B5=0,"Perpendicular","Not perpendicular") |
| a1b2 − a2b1 | =B1*B4-B2*B3 |
| Parallel? | =IF(B7=0,"Parallel","Not parallel") |
Table to find the unit vector
| x component a1 | 4 |
| y component a2 | 2 |
| z component a3 (0 for 2D) | 4 |
| Magnitude |a| | =SQRT(B1^2+B2^2+B3^2) |
| Unit vector, x component | =B1/B4 |
| Unit vector, y component | =B2/B4 |
| Unit vector, z component | =B3/B4 |
| Check - magnitude of the unit vector | =SQRT(B5^2+B6^2+B7^2) |
After pasting, the upper rows (components and coefficients) are the inputs, and the lower rows show the calculated results.
"^" is a power, SQRT is the square root, ACOS is the inverse cosine (a function that finds the angle from a cosine value), and DEGREES converts radians to degrees.
Tables 1 to 4 use the 3D vectors a = (4, 2, 4) and b = (−1, 2, 2). The magnitude |a| is 6, the sum is (3, 4, 6), the difference is (5, 0, 2), 2a is (8, 4, 8), and the dot product is 8. For 2D vectors, set the z component cells to 0.
Table 5 finds the angle from that dot product of 8 and the magnitudes 6 and 3. cosθ is 0.4444…, and the angle is about 63.61 degrees (about 1.1102 radians).
Table 6 uses the 2D vectors a = (4, 2) and b = (−1, 2). The dot product is 0, so it shows "Perpendicular", and a1b2 − a2b1 is 10, not 0, so it shows "Not parallel".
Table 7 finds the unit vector for the same a = (4, 2, 4) as table 1. Each component is divided by the magnitude 6, giving (0.6667, 0.3333, 0.6667), and the bottom row confirms that its magnitude is exactly 1.
"^" is a power, SQRT is the square root, ACOS is the inverse cosine (a function that finds the angle from a cosine value), and DEGREES converts radians to degrees.
Tables 1 to 4 use the 3D vectors a = (4, 2, 4) and b = (−1, 2, 2). The magnitude |a| is 6, the sum is (3, 4, 6), the difference is (5, 0, 2), 2a is (8, 4, 8), and the dot product is 8. For 2D vectors, set the z component cells to 0.
Table 5 finds the angle from that dot product of 8 and the magnitudes 6 and 3. cosθ is 0.4444…, and the angle is about 63.61 degrees (about 1.1102 radians).
Table 6 uses the 2D vectors a = (4, 2) and b = (−1, 2). The dot product is 0, so it shows "Perpendicular", and a1b2 − a2b1 is 10, not 0, so it shows "Not parallel".
Table 7 finds the unit vector for the same a = (4, 2, 4) as table 1. Each component is divided by the magnitude 6, giving (0.6667, 0.3333, 0.6667), and the bottom row confirms that its magnitude is exactly 1.
How to calculate it in Google Sheets
Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.
Table to find the magnitude of a vector
| x component a1 | 4 |
| y component a2 | 2 |
| z component a3 (0 for 2D) | 4 |
| Magnitude |a| | =SQRT(B1^2+B2^2+B3^2) |
Table to find the sum and difference of vectors
| a, x component a1 | 4 |
| a, y component a2 | 2 |
| a, z component a3 | 4 |
| b, x component b1 | -1 |
| b, y component b2 | 2 |
| b, z component b3 | 2 |
| Sum, x component | =B1+B4 |
| Sum, y component | =B2+B5 |
| Sum, z component | =B3+B6 |
| Difference, x component | =B1-B4 |
| Difference, y component | =B2-B5 |
| Difference, z component | =B3-B6 |
Table to find a scalar multiple of a vector
| Scalar k | 2 |
| x component a1 | 4 |
| y component a2 | 2 |
| z component a3 | 4 |
| k times, x component | =B1*B2 |
| k times, y component | =B1*B3 |
| k times, z component | =B1*B4 |
Table to find the dot product
| a, x component a1 | 4 |
| a, y component a2 | 2 |
| a, z component a3 | 4 |
| b, x component b1 | -1 |
| b, y component b2 | 2 |
| b, z component b3 | 2 |
| Dot product a·b | =B1*B4+B2*B5+B3*B6 |
Table to find the angle between two vectors
| Dot product a·b | 8 |
| Magnitude |a| | 6 |
| Magnitude |b| | 3 |
| cosθ | =B1/(B2*B3) |
| Angle θ (degrees) | =DEGREES(ACOS(B4)) |
| Angle θ (radians) | =ACOS(B4) |
Table to check the conditions for perpendicular and parallel (2D)
| a, x component a1 | 4 |
| a, y component a2 | 2 |
| b, x component b1 | -1 |
| b, y component b2 | 2 |
| Dot product a·b | =B1*B3+B2*B4 |
| Perpendicular? | =IF(B5=0,"Perpendicular","Not perpendicular") |
| a1b2 − a2b1 | =B1*B4-B2*B3 |
| Parallel? | =IF(B7=0,"Parallel","Not parallel") |
Table to find the unit vector
| x component a1 | 4 |
| y component a2 | 2 |
| z component a3 (0 for 2D) | 4 |
| Magnitude |a| | =SQRT(B1^2+B2^2+B3^2) |
| Unit vector, x component | =B1/B4 |
| Unit vector, y component | =B2/B4 |
| Unit vector, z component | =B3/B4 |
| Check - magnitude of the unit vector | =SQRT(B5^2+B6^2+B7^2) |
The same formulas as in Excel work as is (SQRT, ACOS, DEGREES and IF have the same names in both). Copy the whole table, paste it into cell A1, and replace the components with your own numbers.
Google Sheets also has a function called SUMPRODUCT. Give it the two ranges of components, as in "=SUMPRODUCT(B1:B3,B4:B6)", and it finds the dot product in a single formula.
Google Sheets also has a function called SUMPRODUCT. Give it the two ranges of components, as in "=SUMPRODUCT(B1:B3,B4:B6)", and it finds the dot product in a single formula.
How to calculate it in Python
import math
from fractions import Fraction
# Vector components (a fraction such as 3/4 can be written Fraction(3, 4); for 2D, use two items)
a = [Fraction(4), Fraction(2), Fraction(4)]
b = [Fraction(-1), Fraction(2), Fraction(2)]
k, l = Fraction(2), Fraction(3)
# Sum, difference and linear combination (component by component)
vector_sum = [x + y for x, y in zip(a, b)]
vector_diff = [x - y for x, y in zip(a, b)]
combination = [k * x + l * y for x, y in zip(a, b)]
# Dot product, magnitudes and angle
dot = sum(x * y for x, y in zip(a, b))
size_a = math.sqrt(sum(x * x for x in a))
size_b = math.sqrt(sum(x * x for x in b))
cos_theta = dot / (size_a * size_b)
theta_degrees = math.degrees(math.acos(cos_theta))
print(f"Sum: {vector_sum}")
print(f"Difference: {vector_diff}")
print(f"ka + lb: {combination}")
print(f"Dot product: {dot}")
print(f"Magnitude |a|: {size_a}, |b|: {size_b}")
print(f"cos(theta): {float(cos_theta)}, angle: {theta_degrees} degrees")
# Unit vector and perpendicular check
unit_a = [float(x) / size_a for x in a]
print(f"Unit vector in the direction of a: {unit_a}")
print("Perpendicular" if dot == 0 else "Not perpendicular")
It uses the standard library only. Keeping the components as Fraction (fractions) makes the sum, difference and dot product exact, with no rounding error. The magnitudes and the angle involve square roots, so they are calculated as decimals with the math module. This example uses a = (4, 2, 4) and b = (−1, 2, 2), and prints the dot product 8, the magnitudes 6 and 3, and an angle of about 63.61 degrees. Change the components and run it.
How to write it in LaTeX and other math languages (copy and paste)
Magnitude of a vector (from its components)
|a| = √(a₁² + a₂²)
\left|\vec{a}\right| = \sqrt{a_1^{2} + a_2^{2}}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mrow><mo>|</mo><mover><mi>a</mi><mo>→</mo></mover><mo>|</mo></mrow>
<mo>=</mo>
<msqrt>
<mrow>
<msup><msub><mi>a</mi><mn>1</mn></msub><mn>2</mn></msup>
<mo>+</mo>
<msup><msub><mi>a</mi><mn>2</mn></msub><mn>2</mn></msup>
</mrow>
</msqrt>
</mrow>
</math>
abs(vec a) = sqrt(a_1^2 + a_2^2)
Norm[{a1, a2}]
LinearAlgebra:-Norm(Vector([a1, a2]), 2);
norm([a1 a2])
|a| = √(a_1^2 + a_2^2)
Sum and difference of vectors (add or subtract component by component)
a + b = (a₁ + b₁, a₂ + b₂)
\vec{a} + \vec{b} = (a_1 + b_1,\ a_2 + b_2)
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mover><mi>a</mi><mo>→</mo></mover>
<mo>+</mo>
<mover><mi>b</mi><mo>→</mo></mover>
<mo>=</mo>
<mo>(</mo>
<msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><msub><mi>b</mi><mn>1</mn></msub>
<mo>,</mo>
<msub><mi>a</mi><mn>2</mn></msub><mo>+</mo><msub><mi>b</mi><mn>2</mn></msub>
<mo>)</mo>
</mrow>
</math>
vec a + vec b = (a_1 + b_1, a_2 + b_2)
{a1, a2} + {b1, b2}
Vector([a1, a2]) + Vector([b1, b2]);
[a1 a2] + [b1 b2]
a + b = (a_1 + b_1, a_2 + b_2)
Scalar multiple of a vector
ka = (ka₁, ka₂)
k\vec{a} = (k a_1,\ k a_2)
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>k</mi><mover><mi>a</mi><mo>→</mo></mover>
<mo>=</mo>
<mo>(</mo>
<mi>k</mi><msub><mi>a</mi><mn>1</mn></msub>
<mo>,</mo>
<mi>k</mi><msub><mi>a</mi><mn>2</mn></msub>
<mo>)</mo>
</mrow>
</math>
k vec a = (k a_1, k a_2)
k {a1, a2}
k * Vector([a1, a2]);
k * [a1 a2]
ka = (k a_1, k a_2)
Dot product (a calculation that turns two vectors into a number)
a ⋅ b = |a||b|cosθ = a₁b₁ + a₂b₂
\vec{a}\cdot\vec{b} = \left|\vec{a}\right|\left|\vec{b}\right|\cos\theta = a_1 b_1 + a_2 b_2
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mover><mi>a</mi><mo>→</mo></mover>
<mo>⋅</mo>
<mover><mi>b</mi><mo>→</mo></mover>
<mo>=</mo>
<msub><mi>a</mi><mn>1</mn></msub><msub><mi>b</mi><mn>1</mn></msub>
<mo>+</mo>
<msub><mi>a</mi><mn>2</mn></msub><msub><mi>b</mi><mn>2</mn></msub>
</mrow>
</math>
vec a * vec b = abs(vec a) abs(vec b) cos theta = a_1 b_1 + a_2 b_2
Dot[{a1, a2}, {b1, b2}]
DotProduct(Vector([a1, a2]), Vector([b1, b2]));
dot([a1 a2], [b1 b2])
a ⋅ b = |a||b| cos θ = a_1 b_1 + a_2 b_2
Angle between two vectors
cos θ = (a ⋅ b) / (|a||b|)
\cos\theta = \frac{\vec{a}\cdot\vec{b}}{\left|\vec{a}\right|\left|\vec{b}\right|}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>cos</mi><mi>θ</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mover><mi>a</mi><mo>→</mo></mover>
<mo>⋅</mo>
<mover><mi>b</mi><mo>→</mo></mover>
</mrow>
<mrow>
<mo>|</mo><mover><mi>a</mi><mo>→</mo></mover><mo>|</mo>
<mo>|</mo><mover><mi>b</mi><mo>→</mo></mover><mo>|</mo>
</mrow>
</mfrac>
</mrow>
</math>
cos theta = (vec a * vec b) / (abs(vec a) abs(vec b))
ArcCos[Dot[a, b]/(Norm[a] Norm[b])]
VectorAngle(Vector([a1, a2]), Vector([b1, b2]));
acos(dot(a, b) / (norm(a) * norm(b)))
cos θ = (a ⋅ b)/(|a||b|)
Conditions for perpendicular and parallel vectors
a⊥b ⟺ a ⋅ b = 0, a∥b ⟺ b = ka
\vec{a}\perp\vec{b} \iff \vec{a}\cdot\vec{b}=0,\qquad \vec{a}\parallel\vec{b} \iff \vec{b}=k\vec{a}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mover><mi>a</mi><mo>→</mo></mover>
<mo>⊥</mo>
<mover><mi>b</mi><mo>→</mo></mover>
<mo>⟺</mo>
<mover><mi>a</mi><mo>→</mo></mover>
<mo>⋅</mo>
<mover><mi>b</mi><mo>→</mo></mover>
<mo>=</mo>
<mn>0</mn>
</mrow>
</math>
vec a _|_ vec b <=> vec a * vec b = 0
Dot[a, b] == 0
evalb(DotProduct(a, b) = 0);
dot(a, b) == 0
a ⊥ b ⟺ a ⋅ b = 0
Unit vector (keeping only the direction)
a / |a| = (a₁/|a|, a₂/|a|)
\frac{\vec{a}}{\left|\vec{a}\right|} = \left(\frac{a_1}{\left|\vec{a}\right|},\ \frac{a_2}{\left|\vec{a}\right|}\right)
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mfrac>
<mover><mi>a</mi><mo>→</mo></mover>
<mrow><mo>|</mo><mover><mi>a</mi><mo>→</mo></mover><mo>|</mo></mrow>
</mfrac>
<mo>=</mo>
<mo>(</mo>
<mfrac>
<msub><mi>a</mi><mn>1</mn></msub>
<mrow><mo>|</mo><mover><mi>a</mi><mo>→</mo></mover><mo>|</mo></mrow>
</mfrac>
<mo>,</mo>
<mfrac>
<msub><mi>a</mi><mn>2</mn></msub>
<mrow><mo>|</mo><mover><mi>a</mi><mo>→</mo></mover><mo>|</mo></mrow>
</mfrac>
<mo>)</mo>
</mrow>
</math>
vec a / abs(vec a) = (a_1/abs(vec a), a_2/abs(vec a))
Normalize[{a1, a2}]
LinearAlgebra:-Normalize(Vector([a1, a2]), 2);
[a1 a2] / norm([a1 a2])
a/|a| = (a_1/|a|, a_2/|a|)
How to have ChatGPT do the calculation
You are an assistant for math calculations (vectors). Do the following calculations by actually running Python code, and base your answer only on the numbers from the output (do not answer from mental math or guesses). For the 3D vectors a = (4, 2, 4) and b = (−1, 2, 2), find each of the following. 1. The components of the sum a + b, the difference a − b, and 2a + 3b 2. The magnitudes |a| and |b| (both as exact values with simplified radicals and as decimals) 3. The dot product a·b 4. The angle θ between them (in both degrees and radians, with the exact value if it is a special angle) 5. The unit vector in the direction of a 6. Whether a and b are perpendicular or parallel, and why In Python, use the fractions and math modules (and sympy if needed), and show the formulas you used and the numbers from the output.
How to Use
-
1Enter your numbersType the numbers you want to calculate with into the input fields
-
2CalculatePress the "Calculate" button
-
3Check the resultThe result appears on the spot. The same page also explains the idea behind the calculation and the formula
DataChef Features
Easy and Free
Unlimited conversions for free.
No technical knowledge required.
Intuitive and user-friendly operation.
No technical knowledge required.
Intuitive and user-friendly operation.
No Registration Required
Available immediately after access.
Can be used without registering personal information.
Can be used without registering personal information.
Safe and Secure
Fully SSL encrypted communication.
Automatic file deletion by clicking "download".
Automatic file deletion by clicking "download".
Fast
High-speed site access
and rapid file conversion.
and rapid file conversion.
No Watermark
No watermark.
No attribution required.
No attribution required.
Commercial Use Available
Free for commercial use.
No need to contact us for commercial use permission.
No need to contact us for commercial use permission.
Related Features
Cross Product Calculator (Components, Parallelogram Area, Normal Vector)
Calculates the cross product a×b of two 3D vectors (a vector perpendicular to both) and the area of the parallelogram they form
Pythagorean Theorem Calculator (Hypotenuse and Legs of a Right Triangle)
The Pythagorean theorem is the reason behind the magnitude of a vector. Finds the side lengths of a right triangle
Sin Cos Tan Calculator (Trig Ratios and Angles with a Unit Circle)
See how cos values match angles, as used when finding the angle between vectors. Shows exact values and decimals
Ad 
