Prove that the position vectors $4 i + 5 j + 6 k, 5 i + 6 j + 4 k,$ & $6 i + 4 j + 5 k$ form an equilateral triangle.

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Solution

Let position vectors be.
$A = 4 \land i + 5 \land j + 6 \land k$
$B = 5 \land i + 6 \land j + 4 \land k$
$C = 6 \land i + 4 \land j + 5 \land k$
$¯ ¯¯¯¯¯¯ ¯ A B = (5 - 4) \land i + (6 - 5) \land j + (4 - 6) \land k$
$= \land i + \land j - 2 \land k$
$⟹ ∣ ∣ ¯ ¯¯¯¯¯¯ ¯ A B ∣ ∣ = \sqrt{1 + 1 + 2^{2}} = \sqrt{6}$
Similarily, $¯ ¯¯¯¯¯¯ ¯ B C = \land i - 2 \land j + \land k$
$⟹ ∣ ∣ ¯ ¯¯¯¯¯¯ ¯ B C ∣ ∣ = \sqrt{1 + 2^{2} + 1} = \sqrt{6}$
$¯ ¯¯¯¯¯¯ ¯ A C = 2 \land i - \land j - \land k$
$⟹ ∣ ∣ ¯ ¯¯¯¯¯¯ ¯ A C ∣ ∣ = \sqrt{2^{2} + 1^{2} + 1^{2}} = \sqrt{6}$
$∵ ∣ ∣ ¯ ¯¯¯¯¯¯ ¯ A B ∣ ∣ = ∣ ∣ ¯ ¯¯¯¯¯¯ ¯ B C ∣ ∣ = ∣ ∣ ¯ ¯¯¯¯¯¯ ¯ A C ∣ ∣,$
$∴$ All three sides of triangle are equal,
Hence equilateral triangle.

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Prove that the position vectors 4i+5j+6k,5i+6j+4k, & 6i+4j+5k form an equilateral triangle.

Prove that the position vectors $4 i + 5 j + 6 k, 5 i + 6 j + 4 k,$ & $6 i + 4 j + 5 k$ form an equilateral triangle.