Answer to Question #213696 in Analytic Geometry for pkaj

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

Let us determine whether ~u and ~v are orthogonal vectors, make an acute or obtuse angle using dot product of vectors.

(1.1) Since the dot product

~$u\cdot$~$v =< 1, 3, −2 >\cdot< −5, 3, 2 >=1(-5)+3\cdot 3+(-2)\cdot 2=-5+9-4=0,$ we conclude that two vectors are orthogonal.

(1.2) Since the dot product

~$u\cdot$~u =$< 1, −2, 4 >\cdot< 5, 3, 7 >=1\cdot 5+(-2)\cdot 3+4\cdot 8=5-6+32=31>0,$ we conclude that two vectors make an acute angle.