Math Calculators, Lessons and Formulas

It is time to solve your math problem

mathportal.org

Vector calculator

google play badge app store badge

This calculator performs all vector operations in two- and three-dimensional space. You can add, subtract, find length, find vector projections, and find the dot and cross product of two vectors. For each operation, the calculator writes a step-by-step, easy-to-understand explanation of how the work has been done.

Find the magnitude of the vector $ \| \vec{v} \| = \left(\dfrac{ 1 }{ 2 },~\sqrt{ 3 },~5\right) $ .

solution

The magnitude of vector $ \vec{v} $ is $ \| \vec{v} \| = \dfrac{\sqrt{ 113 }}{ 2 } $ .

explanation

To find magnitude of a vector $ v=(a,b,c) $ we use formula $ ||v|| = \sqrt{a^2+b^2+c^2} $

In this example $ a = \frac{ 1 }{ 2 } $ , $ b = \sqrt{ 3 } $ and $ c = 5 $ so:

$$ ||v|| = \sqrt{ \left(\frac{ 1 }{ 2 }\right)^2 + \left( \sqrt{ 3 } \right)^{ 2 } + 5^2} = \sqrt{ \frac{ 1 }{ 4 } + 3 + 25} = \sqrt{ \frac{ 113 }{ 4 } } = \frac{\sqrt{ 113 }}{ 2 }$$

Report an Error!

Script name : vector-calculator

Form values: 1 , 1/2 , r3 , 5 , 4 , -r3 , 10 , dot , g , , , , , , , , , mag , , , ,

Comment (optional)

 
close
Vectors in two dimensions
seven operations on two-dimensional vectors + steps
help ↓↓ examples ↓↓ tutorial ↓↓
vectors in two dimensions
V1(1,-3) V2(5,1/2)
V1(√2,-1/3) V2(√5,0)
Magnitude (length) of V1
V1 · V2 ( dot product )
V1 + V2 V1 - V2
Angle between V1 and V2
Check if V1 and V2 are linearly dependent
Find projection of V1 onto V2
Find approximate solution
Hide steps
thumb_up 1.6K thumb_down

Get Widget Code

Vectors in three dimensions
seven operations on three-dimensional vectors + steps
help ↓↓ examples ↓↓ tutorial ↓↓
vectors in three dimensions
V1=(1,-√3,3/2) V2=(√2, 1, 2/3)
Magnitude (length) of V1
V1 · V2 ( dot product ) V1 ✕ V2 ( cross product )
V1 + V2 V1 - V2
Angle between V1 and V2
Check if V1, V2 and V3 are linearly dependent
Find projection of V1 onto V2
Find approximate solution
Hide steps
thumb_up 1.6K thumb_down

Get Widget Code

working...
Examples
ex 1:
Given vector v1=(8,-4), calculate the magnitude.
ex 2:
Calculate the difference of vectors v1=(3/4,2) and v2=(3,-2).
ex 3:
Calculate the dot product of vectors v1=(-1/4,2/5) and v2=(-5,-5/4)
ex 4:
Find the angle between the vectors v1=(3,5,-7) and v2=(-3,4,-2)
ex 5:
Find the cross product of v1=(-2,2/3,-3) and v2=(4,0,-1/2)
ex 6:
Determine if the following set of vectors is linearly independent: v1=(3,-2,4), v2=(1,-2,3) and v3=(3,2,-1)
Find more worked-out examples in our database of solved problems..
TUTORIAL

Vector operations

In this tutorial, we'll learn how to find: magnitude, dot product, angle between two vectors and cross product of two vectors.

1 : Magnitude

Magnitude is the vector length. The formula for the magnitude of a vector $\vec{v} = (v_1, v_2) $ is:

$$ \| \vec{v} \| = \sqrt{v_1^2 + v_2^2 } $$

Example 01: Find the magnitude of the vector $ \vec{v} = (4, 2) $.

In this example we have $ v_1 = 4 $ and $ v_2 = 2 $ so the magnitude is:

$$ \| \vec{v} \| = \sqrt{4^2 + 2 ^2} = \sqrt{20} = 2\sqrt{5} $$ vector magnitude

Example 02: Find the magnitude of the vector $ \vec{v} = \left(\dfrac{2}{3}, \sqrt{3}, 2\right) $.

Solution

2 : Dot product

The formula for the dot product of vectors $ \vec{v} = (v_1, v_2) $ and $ \vec{w} = (w_1, w_2) $ is

$$ \vec{v} \cdot \vec{w}= v_1 \cdot w_1 + v_2 \cdot w_2 $$

Two vectors are orthogonal to each other if their dot product is equal zero.

Example 03: Calculate the dot product of $ \vec{v} = \left(4, 1 \right) $ and $ \vec{w} = \left(-1, 5 \right) $. Check if the vectors are mutually orthogonal.

To find the dot product we use the component formula:

$$ \begin{aligned} \vec{v} \cdot \vec{w} &= \left(4, 1 \right) \cdot \left(-1, 5 \right) = \\[1 em] &= 4 \cdot (-1) + 1 \cdot 5 = -4 + 5 = 1 \end{aligned} $$ vector dot product

Since the dot product is not equal to zero, we can conclude that vectors ARE NOT orthogonal.

Example 04: Find the dot product of the vectors $ \vec{v_1} = \left(\dfrac{1}{2}, \sqrt{3}, 5 \right) $ and $ \vec{v_2} = \left( 4, -\sqrt{3}, 10 \right) $.

Solution

3 : Angle between two vectors

To find the angle $ \alpha $ between vectors $ \vec{a} $ and $ \vec{b} $, we use the following formula:

$$ \cos \alpha = \dfrac{\vec{a} \cdot \vec{b}}{ \|\vec{a}\| \, \|\vec{b}\|} $$

Note that $ \vec{a} \cdot \vec{b} $ is a dot product while $\|\vec{a}\|$ and $\|\vec{b}\|$ are magnitudes of vectors $ \vec{a} $ and $ \vec{b}$.

Example 05: Find the angle between vectors $ \vec{a} = ( 4, 3) $ and $ \vec{b} = (-2, 2) $.

First we will find the dot product and magnitudes:

$$ \begin{aligned} \vec{a} \cdot \vec{b} &= ( 4, 3) \cdot (-2, 2) = 4 \cdot (-2) + 3 \cdot 2 = -2 \\[1 em] \| \vec{a} \| &= \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \\[1 em] \| \vec{b} \| &= \sqrt{(-2)^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} \end{aligned} $$

Now we'll find the $ \cos \alpha $

$$ \cos \alpha = \dfrac{\vec{a} \cdot \vec{b}}{ \|\vec{a}\| \, \|\vec{b}\|} = \dfrac{-2}{5 \sqrt{8}} \approx -0.1414 $$

The angle $ \alpha $ is:

$$ \alpha = \cos^{-1}(-0.1414) \approx 98^{o} $$ vector angle between two vectors

Example 06: Find the angle between vectors $ \vec{v_1} = \left(2, 1, -4 \right) $ and $ \vec{v_2} = \left( 3, -5, 2 \right) $.

Solution

4 : Cross product

The cross product of vectors $ \vec{v} = (v_1,v_2,v_3) $ and $ \vec{w} = (w_1,w_2,w_3) $ is given by the formula:

$$ \vec{v} \times \vec{w} = \left( v_2 w_3 - v_3 w_2, v_3 w_1 - v_1 w_3, v_1 w_2 - v_2 w_1 \right) $$

Note that the cross product requires both vectors to be three-dimensional.

If the two vectors are parallel than the cross product is equal zero.

Example 07: Find the cross products of the vectors $ \vec{v} = ( -2, 3 , 1) $ and $ \vec{w} = (4, -6, -2) $. Check if the vectors are parallel.

We'll find cross product using above formula

v • w =
= (3×(-2)-1×(-6), 1×4-(-2)×(-2), -2×(-6)-3×4) =
= (0, 0, 0)

Since the cross product is zero, we conclude that the vectors are parallel.

Example 08: Find the cross products of the vectors $ \vec{v_1} = \left(4, 2, -\dfrac{3}{2} \right) $ and $ \vec{v_2} = \left(\dfrac{1}{2}, 0, 2 \right) $.

Solution

Search our database with more than 300 calculators
362 861 664 solved problems
×
ans:
syntax error
C
DEL
ANS
±
(
)
÷
×
7
8
9
4
5
6
+
1
2
3
=
0
.