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Vector projection

Definition. Projection of the vector AB on the axis l is a number equal to the value of the segment A1B1 on axis l, where points A1 and B1 are projections of points A and B on the axis l (Fig. 1).
Projection of the vector on axis
Fig. 1
Definition. The vector projection of a vector a on a nonzero vector b is the orthogonal projection of a onto a straight line parallel to b.

Vector projection - formula

The vector projection of a on b is the unit vector of b by the scalar projection of a on b:

proj ba = a · bb
|b|2

The scalar projection of a on b is the magnitude of the vector projection of a on b.

|proj ba| = a · b
|b|

Examples of tasks

Examples of plane tasks

Example 1. Find the projection of vector a = {1; 2} on vector b = {3; 4}.

Solution:

Calculate dot product of these vectors:

a · b = 1 · 3 + 2 · 4 = 3 + 8 = 11

Calculate the magnitude of vector b:

|b| = √32 + 42 = √9 + 16 = √25 = 5

Calculate vector projection:

proj ba = a · b b = 11{3; 4} ={1.32; 1.76}
|b|225

Calculate scalar projection:

|proj ba| = a · b  = 11 = 2.2
|b|5

Examples of spatial tasks

Example 2. Find the projection of vector a = {1; 4; 0} on vector b = {4; 2; 4}.

Solution:

Calculate dot product of these vectors:

a · b = 1 · 4 + 4 · 2 + 0 · 4 = 4 + 8 + 0 = 12

Calculate the magnitude of vector b:

|b| = √42 + 22 + 42 = √16 + 4 + 16 = √36 = 6

Calculate vector projection:

proj ba = a · b b = 12{4; 2; 4} = {424}
|b|236333

Calculate scalar projection:

|proj ba| = a · b  = 12 = 2
|b|6

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