Question

What is the projection of <2,−4,3> onto <1,2,2>?

Answer 1

Example

The dot product of a=<1,3,-2> and b=<-2,4,-1> is

Using the (**)we see that

which implies theta=45.6 degrees.

An important use of the dot product is to test whether or not two vectors are orthogonal. Two vectors are orthogonal if the angle between them is 90 degrees. Thus, using (**) we see that the dot product of two orthogonal vectors is zero. Conversely, the only way the dot product can be zero is if the angle between the two vectors is 90 degrees (or trivially if one or both of the vectors is the zero vector). Thus, two non-zero vectors have dot product zero if and only if they are orthogonal.

Example

<1,-1,3> and <3,3,0> are orthogonal since the dot product is 1(3)+(-1)(3)+3(0)=0.

Projections

One important use of dot products is in projections. The scalar projection of b onto a is the length of the segment AB shown in the figure below. The vector projection of b onto a is the vector with this length that begins at the point A points in the same direction (or opposite direction if the scalar projection is negative) as a.

Thus, mathematically, the scalar projection of b onto a is |b|cos(theta) (where theta is the angle between a and b) which from (*) is given by

This quantity is also called the component of b in the a direction (hence the notation comp). And, the vector projection is merely the unit vector a/|a| times the scalar projection of b onto a:

Thus, the scalar projection of b onto a is the magnitude of the vector projection of b onto a.