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Vector Addition

Consider a AB...

Question

Consider a $△ A B C,$ whose vertices are $A (- 8, 2, 4), B (4, - 2, 3)$ and $C (1, 3, 2)$ . Then the value of $A B^{2} - 2 B C^{2} - A C^{2} =$

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Solution

Given : vertices of triangle $A (- 8, 2, 4), B (4, - 2, 3)$ and $C (1, 3, 2)$
$\Rightarrow A B^{2} = (4 + 8)^{2} + (- 2 - 2)^{2} + (3 - 4)^{2} \Rightarrow A B^{2} = 161,$
$B C^{2} = (4 - 1)^{2} + (3 + 2)^{2} + (2 - 3)^{2} = 35$ and
$A C^{2} = (1 + 8)^{2} + (3 - 2)^{2} + (2 - 4)^{2} = 86$
$∴ A B^{2} - 2 B C^{2} - A C^{2} = 161 - 2 \times 35 - 86 = 5$

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