Vector Projection Calculator

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

Vector a
Vector b
Vector a
Vector b

Projection of a onto b (projba):

 

Steps to Solve

Use the Vector Projection Formula

projba = a · b / |b|²b

Substitute Values and Solve

Enter vectors a & b above to see the solution here

Learn how we calculated this below


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How to Project One Vector Onto Another

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

A vector projection is denoted projba, which reads 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:

projba = 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.

projba = (8 · 0.5046, 3 · 0.5046, 6 · 0.5046)
projba = (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.