Is the triangle formed by the vectors $3 i + 5 j + 2 k, 2 i - 3 j - 5 k a n d - 5 i - 2 j + 3 k$ equilateral?

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Solution

$| 3 ˆ i + 5 ˆ j + 2 ˆ k | = \sqrt{3^{2} + 5^{2} + 2^{2}} = \sqrt{38}$
$| 2 ˆ i - 3 ˆ j - 5 ˆ k | = \sqrt{2^{2} + (- 3)^{2} + (- 5)^{2}} = \sqrt{38}$
$| - 5 ˆ i - 2 ˆ j + 3 ˆ k | = \sqrt{(- 5)^{2} + (- 2)^{2} + 3^{2}} = \sqrt{38}$
Hence, the triangle formed by the given vectors is equilateral.

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