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Question
The vertices of ΔABC are A(7, 2, 1), 5(6, 0, 3), and C(4, 2, 4). Find ∠ABC
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Solution
∠ABC = Angle between `vec"AB"` and `vec"BC"`
`vec"a"*vec"b" = |vec"a"||vec"b"| cos theta`
`vec"AB" = vec"OB" - vec"OA"`
= `(6 - 7)hat"i" + (0 - 2)hat"j" + (3 - 1)hat"k"`
= `- hat"i" - 2hat"j" + 2hat"k"`
`vec"BC" = vec"OC" - vec"OB"`
= `(4 - 6)hat"i" + (2 - 0)hat"j" + (4 - 3)hat"k"`
= `-2hat"i" + 2hat"j" + hat"k"`
Now `vec"AB"*vec"BC" = (-hat"i" - 2hat"j" + 2hat"k")*(-2hat"i" + 2hat"j" + hat"k")`
= 2 – 4 + 2
= 0
cos θ = 0
θ = `pi/2`
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