Graphs of Functions and Derivatives 133Parallel and Perpendicular Vectors
Ifr 2 =Cr 1 , thenr 1 andr 2 are parallel.
Ifr 1 ·r 2 =0, thenr 1 andr 2 are perpendicular or orthogonal.
The angle between two vectors can be found by cosθ=
r 1 ·r 2
‖r 1 ‖·‖r 2 ‖.
Example 1
Givenr 1 =〈4,− 7 〉,r 2 =〈−3,− 2 〉andr 2 =〈−1, 5〉, find 3r 1 − 5 r 2 + 2 r 3.
3 r 1 − 5 r 2 − 2 r 3 = 3 〈4,− 7 〉 − 5 〈−3,− 2 〉 + 2 〈−1, 5〉 = 〈12,− 21 〉 − 〈−15,− 10 〉 +
〈−2, 10〉=〈27,− 11 〉−〈−2, 10〉=〈29,− 21 〉.Example 2
Determine whether the vectorsr 1 =〈4,− 7 〉andr 2 =〈−3,− 2 〉are orthogonal. If the vectors
are not orthogonal, approximate the angle between them.Step 1: Find the dot productr 1 ·r 2 =4(−3)+(−7)(−2)=2. Since the dot product is not
equal to zero, the vectors are not orthogonal.
Step 2: Ifθis the angle between the vectors, then cosθ=
r 1 ·r 2
‖r 1 ‖·‖r 2 ‖. The dot product is
2,‖r 1 ‖=√
65, and‖r 2 ‖=√
13, so cosθ=2
√
65 ·√
13=
2
√
5
65
≈ 0 .0688 and
θ≈ 1 .5019 radians.7.6 Rapid Review
- Iff′(x)=x^2 −4, find the intervals where fis decreasing. (See Figure 7.6-1.)
Figure 7.6-1Answer:Sincef′(x)<0if− 2 <x<2,f is decreasing on (−2, 2).