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Applications of Dot Product

Show that vec...

Question

Show that vectors $∣ ∣ ¯ b ∣ ∣ ¯ a + ∣ ∣ ¯ a ∣ ∣ ¯ b$ and $∣ ∣ ¯ b ∣ ∣ ¯ a - ∣ ∣ ¯ a ∣ ∣ ¯ b$ are orthogonal.

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Solution

$\begin{matrix} \to c = ∣ ∣ ∣ \to b ∣ ∣ ∣ \to a + ∣ ∣ \to a ∣ ∣ \to b \to d = ∣ ∣ ∣ \to b ∣ ∣ ∣ \to a - ∣ ∣ \to a ∣ ∣ \to b \to c . \to d = (∣ ∣ ∣ \to b ∣ ∣ ∣ \to a + ∣ ∣ \to a ∣ ∣ \to b) . (∣ ∣ ∣ \to b ∣ ∣ ∣ \to a - ∣ ∣ \to a ∣ ∣ \to b) = {∣ ∣ ∣ \to b ∣ ∣ ∣}^{2} + {∣ ∣ \to a ∣ ∣}^{2} - ∣ ∣ ∣ \to b ∣ ∣ ∣ ∣ ∣ \to a ∣ ∣ \to a . \to b + ∣ ∣ \to a ∣ ∣ ∣ ∣ ∣ \to b ∣ ∣ ∣ \to a . \to b - {∣ ∣ \to a ∣ ∣}^{2} {∣ ∣ ∣ \to b ∣ ∣ ∣}^{2} = 0 \to c & \to d a r e o r t h o g o n a l . \end{matrix}$

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