Vector Projection Calculator

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

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_ba, 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_ba = 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_ba = (8 · 0.5046, 3 · 0.5046, 6 · 0.5046)
proj_ba = (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.