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Applications of Dot Product

If a⃗,b⃗,c⃗...

Question

If $\to a, \to b, \to c$ are mutually perpendicular vectors of equal magnitude, then $\to a + \to b + \to c$ is equally inclined to

A

\to a

only

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B

\to a, \to c

only

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C

All

\to a, \to b, \to c

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D

none of these

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Solution

The correct option is D All $\to a, \to b, \to c$
Given

$\to a . \to b = \to b . \to c = \to c . \to a = 0$
And
$| \to a | = | \to b | = | \to c | = k$ (say)

$\to a . (\to a + \to b + \to c) = | \to a | | \to a + \to b + \to c | cos α$
Where $α$ is the angle between $\to a$ and $\to a + \to b + \to c$

$\to a . \to a + \to a . \to b + \to a . \to c = k | \to a + \to b + \to c | cos α$

$| \to a |^{2} + \to a . \to b + \to a . \to c = k | \to a + \to b + \to c | cos α$

$k^{2} = k | \to a + \to b + \to c | cos α$

$cos α = \frac{k}{| \to a + \to b + \to c |}$ ----- (i)

$\to b . (\to a + \to b + \to c) = | \to b | | \to a + \to b + \to c | cos β$
Where $β$ is the angle between $\to b$ and $(\to a + \to b + \to c)$

$\to b . \to a + \to b . \to b + \to b . \to c = k | \to a + \to b + \to c | cos β$

$| \to b |^{2} = k | \to a + \to b + \to c | cos β$

$k^{2} = k | \to a + \to b + \to c | cos β$

$cos β = \frac{k}{| \to a + \to b + \to c |}$ ----- (ii)
Similarly,
$\to c . (\to a + \to b + \to c) = | \to c | | \to a + \to b + \to c | cos γ$
Where $γ$ is the angle between $\to c$ and $(\to a + \to b + \to c)$

$\to c . \to a + \to c . \to b + \to c . \to c = k | \to a + \to b + \to c | cos γ$

$| \to c |^{2} = k | \to a + \to b + \to c | cos γ$

$k^{2} = k | \to a + \to b + \to c | cos γ$

$cos γ = \frac{k}{| \to a + \to b + \to c |}$ ----- (iii)
From (i),(ii), and (iii)

$cos α = cos β = cos γ$

Hence $\to a + \to b + \to c$ is equally inclined to all $\to a, \to b$ and $\to c$

Hence, Option $C$ is the correct answer.

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