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If A,B, C have position vectors (0,1,1),...

If `A`,`B`, `C` have position vectors `(0,1,1)`,`(3,1,5)`,`(0,3,3)`, respectively, then show that `Delta ABC` is right angled at `C`.

Text Solution

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Here, Position vector of `A = hatj+hatk`
Position vector of `B = 3hati+hatj+5hatk`
Position vector of `C = 3hatj+3hatk`
`vec(AC) = (3hatj+3hatk)-(hatj+hatk) = 2hatj+2hatk`
`vec(BC) = (3hatj+3hatk) - (3hati+hatj+5hatk) = -3hati+2hatj-2hatk`
`vec(AC).vec(BC) = 0+2(2)+2(-2) = 4-4 = 0`
As, dot product of AC and BC is `0`, it mean they are perpendicular.
`:. Delta ABC` is a right angled triangle with `/_C =90^@`
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