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Applications of Dot Product

If a⃗, b⃗ a...

Question

If $\to a, \to b$ and $\to c$ are three mutually perpendicular vectors, then the vector which is equally inclined to these vectors is

A

\to a + \to b + \to c

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B

\frac{\to a}{| \to a |} + \frac{\to b}{| \to b |} + \frac{\to c}{| \to c |}

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C

\frac{\to a}{| \to a |^{2}} + \frac{\to b}{| \to b |^{2}} + \frac{\to c}{| \to c |^{2}}

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D

| \to a | \to a - | \to b | \to b + | \to c | \to c

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Solution

The correct option is A $\frac{\to a}{| \to a |} + \frac{\to b}{| \to b |} + \frac{\to c}{| \to c |}$
Let $\to α = \frac{\to a}{| \to a |} + \frac{\to b}{| \to b |} + \frac{\to c}{| \to c |}$
Since $\to a, \to b$ and $\to c$ are mutually perpendicular vectors, if $\to α$ makes angles $θ, ϕ, Ψ$ with $\to a, \to b$ and $\to c$ , respectively, then
$\to α \cdot \to a = \frac{\to a \cdot \to a}{| \to a |}$
or $| \to α | \cdot | \to a | cos θ = | \to a |$
or $cos θ = \frac{1}{| \to α |}$
Similarly, $cos ϕ = \frac{1}{| \to α |}, cos Ψ = \frac{1}{| \to α |}$
$∴ θ = ϕ = Ψ$

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is orthogonal to

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