Answer to Question #213695 in Analytic Geometry for anuj

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

$u=(1,3,-2), v=(-5,3,2)\\ uv=-5+9-4=0\\$

u orthogonal v

1.2

$u=(1,-2,4), v=(5,3,7)\\ uv=5-6+28=27>0$

u is not orthogonal v

$||u||=\sqrt{1+4+16}=\sqrt{21}\\ ||v||=\sqrt{25+9+49}=\sqrt{83}\\ \cos\phi=\frac{uv}{||u||\cdot||v||}=\frac{27}{\sqrt{21}\cdot\sqrt{83}}>0$

acute angle