Answer to Question #213697 in Calculus for amit

Question

Determine whether ~u and ~v are orthogonal vectors, make an acute or obtuse angle: (1.1) ~u =< 1, 3, −2 >, ~v =< −5, 3, 2 >. (2) (1.2) ~u =< 1, −2, 4 >, ~v =< 5, 3, 7 >.

Answer 1

(1.1)

$$u\cdot v=1(-5)+3(3)-2(2)=0$$

The vectors $u$ and $v$ are orthogonal.

(1.2)

$$u\cdot v=1(5)-2(3)+4(7)=27>0$$

$$=>\cos\theta>0=>0\degree<\theta<90\degree$$

The vectors $u$ and $v$ make an acute angle.