Finding the projection of a vector: formula, examples

In this publication, we will consider what the projection of a vector onto an axis or onto another vector is, and we will give a formula with which you can find the value of this projection. We will also analyze examples of solving problems on this topic.

Finding the projection of a vector

Vector projection AB on the axis l is the number that equals the segment A₁B₁. points A₁ и B₁ are projections of points A и B on the axis l.

Vector projection a to the direction of the vector b is a number that is equal to the projection a on an axis passing through b.

Formula for finding the projection of a vector onto a vector

Calculate projection a to direction b can be as follows:

Examples of tasks

Task 1

Let’s find the projection of the vector a = {3; 5} on b = {2; 8}.

Decision:

1. First, we calculate the scalar product of the given vectors:

a · b = 3 · 2 + 5 · 8 = 46

2. Now calculate the length (modulus) b:

3. It remains only to use the formula above to find the projection of the vector:

Task 2

Calculate the projection of the vector a = {4; -two; 7} on b = {11; 3; 6}.

Decision:

Alternately, we perform the same actions as in the example discussed above.

a · b = 4 · 11 + (-7) · 3 + 5 · 6 = 53