CK-12-Pre-Calculus Concepts

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

u·v=<u 1 ,u 2 >·<v 1 ,v 2 >

=u 1 v 1 +u 2 v 2

=v 1 u 1 +v 2 u 2

=<v 1 ,v 2 >·<u 1 ,u 2 >

&=v·u

Example B
Find the dot product between the following vectors:< 3 , 1 >·< 5 ,− 4 >
Solution: < 3 , 1 >·< 5 ,− 4 >= 3 · 5 + 1 ·(− 4 ) = 15 − 4 = 11
Example C
Prove the angle between two vectors formula:
cosθ=|uu||·vv|
Solution: Start with the law of cosines.

|u−v|^2 =|v|^2 +|u|^2 − 2 |v||u|cosθ

(u−v)·(u−v) =

u·u− 2 u·v+v·v=

|u|^2 − 2 u·v+|v|^2 =

− 2 u·v=− 2 |v||u|cosθ

u·v

|u||v|=cosθ

Concept Problem Revisited