Answer to Question #210355 in Analytic Geometry for Ree

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 >.

(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