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 >.
1.1
u=(1,3,−2),v=(−5,3,2)uv=−5+9−4=0u=(1,3,-2), v=(-5,3,2)\\ uv=-5+9-4=0\\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>0u=(1,-2,4), v=(5,3,7)\\ uv=5-6+28=27>0u=(1,−2,4),v=(5,3,7)uv=5−6+28=27>0
u is not orthogonal v
∣∣u∣∣=1+4+16=21∣∣v∣∣=25+9+49=83cosϕ=uv∣∣u∣∣⋅∣∣v∣∣=2721⋅83>0||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∣∣u∣∣=1+4+16=21∣∣v∣∣=25+9+49=83cosϕ=∣∣u∣∣⋅∣∣v∣∣uv=21⋅8327>0
acute angle
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