# Vector Projection Calculator

Project one vector onto another using the calculator below. See the steps to solve along with the solution below.

## Projection of a onto b (proj_{b}a):

_{b}a):

### Steps to Solve

#### Use the Vector Projection Formula

proj_{b}a = a · b / |b|²b

#### Substitute Values and Solve

Enter vectors a & b above to see the solution here

_{b}a):

### Steps to Solve

#### Use the Vector Projection Formula

proj_{b}a = a · b / |b|²b

#### Substitute Values and Solve

Enter vectors a & b above to see the solution here

## On this page:

## How to Project One Vector Onto Another

You can use vector projection to determine how much of one vector goes in the direction of another vector. When projecting a vector onto another vector, the result is a vector that is parallel to the second vector.

A vector projection is denoted *proj _{b}a*, which reads as the projection of vector

*a*onto

*b*.

### Vector Projection Formula

In order to project one vector onto another, you need to use a formula. The vector projection formula is:

proj_{b}a = a · b|b|²b

Where:

*a · b*= the dot product*|b|*= magnitude of vector*b*

Keep reading to see each step to use this formula.

### Vector Projection Example

To use the projection formula, you’ll need to follow a few steps. Follow along, and we’ll go through how to project the vector (2, 5, 4) onto vector (8, 3, 6).

#### Step One: Calculate the Dot Product

First, use the dot product formula to calculate the dot product *a · b*.

a · b = (2 · 8) + (5 · 3) + (4 · 6)

a · b = 16 + 15 + 24

a · b = 55

#### Step Two: Calculate the Magnitude of b

Next, use the vector magnitude formula to calculate the magnitude *b*.

|b|= 8² + 3² + 6²

|b|= 64 + 9 + 36

|b| = 109

#### Step Three: Calculate the Projection Factor

Then, calculate the projection factor by dividing the dot product by the square root of the magnitude of *b* squared.

projection factor = 55 / √109²

projection factor = 55 / 109

projection factor = 0.5046

#### Step Four: Multiply Vector b by the Projection Factor

And finally, multiply each component of vector *b* by the projection factor to complete the projection.

proj_{b}a = (8 · 0.5046, 3 · 0.5046, 6 · 0.5046)

proj_{b}a = (4.037, 1.514, 3.028)

So, projecting vector *a* onto *b* results in the vector *(4.037, 1.514, 3.028)*.

You might also be interested in our vector addition and vector subtraction calculators.