Vector Projection Calculator
Project one vector onto another using the calculator below. See the steps to solve along with the solution below.
Results: 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
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
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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 projba, 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:
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.