Answer to Question #212415 in Analytic Geometry for Janien

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 use the dot products of vectors.

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

(1.2) Since $\overline u \cdot \overline v=< 1, −2, 4 >\cdot< 5, 3, 7 >=5-6+28=27>0,$ we conclude that the vectors

make an acute angle.