CK-12-Pre-Calculus Concepts

http://www.ck12.org Chapter 7. Vectors

The definition of vector projection for the indicated red vector is the calledpro juv. When you readpro juvyou
should say “the vector projection ofvontou.” This implies that the new vector is going in the direction of
u. Conceptually, this means that if someone was pulling the box at an angle and strength of vectorvthen some
of their energy would be wasted pulling the box up and some of the energy would actually contribute to pulling the
box horizontally.
The definition of scalar projection is simply the length of the vector projection. When the scalar projection is
positive it means that the angle between the two vectors is less than 90◦. When the scalar projection is negative it
means that the two vectors are heading in opposite directions.
The vector projection formula can be written two ways. The version on the left is most simplified, but the version
on the right makes the most sense conceptually. The proof is demonstrated in Example A.

pro juv=

(v·u

|u|^2

)

u=

(v·u

|u|

)u
|u|
Example A
Prove the vector projection formula.
Solution: Given two vectorsu,v, what ispro juv?

First note that the projected vector in red will go in the direction ofu. This means that it will be a product of the
unit vector|uu|and the length of the red vector (the scalar projection). In order to find the scalar projection, note the
right triangle, the unknown angleθbetween the two vectors and the cosine ratio.
cosθ=scalar pro jection|v|
Recall that cosθ=|uu||·vv|. Now just substitute and simplify to find the length of the scalar projection.

cosθ=scalar pro jection|v|

u·v

|u||v|=

scalar pro jection

|v|

u·v

|u| =scalar pro jection

Now you have the length of the vector projection and the direction you want it to go:

pro juv=

(u·v

|u|

)u
|u|
Example B
Find the scalar projection of vectorv=< 3 , 4 >onto vectoru=< 5 ,− 12 >.
Solution: As noted in Example A, the scalar projection is the magnitude of the vector projection. This was shown
to be

(u·v

|u|

)

whereuis the vector being projected onto.

u|u·v|=<^5 ,−^1213 >·<^3 ,^4 >= 1513 − (^48) =− (^3313)