If a- b- c- are three mutually perpendicular vectors of equal magnitude- then the angle - which a- b-c- makes with any one of three given vectors is given by

It is given that, |a⃗|=|b⃗|=|c⃗|=λ (say) and a⃗, b⃗, c⃗ are mutually perpendicular vectors. Therefore,|a⃗+b⃗+c⃗|=√(3)λ Let θ be the angle which a⃗+b⃗+c⃗ makes w ...

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Standard XIII

Mathematics

Angle between Two Vectors

If a⃗, b⃗, c⃗...

Question

If $\to a, \to b, \to c$ are three mutually perpendicular vectors of equal magnitude, then the angle $θ$ which $\to a + \to b + \to c$ makes with any one of three given vectors is given by

c o s^{- 1} (\frac{1}{\sqrt{3}})

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c o s^{- 1} (\frac{1}{3})

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c o s^{- 1} (\frac{2}{\sqrt{3}})

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N o n e o f t h e s e

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Solution

The correct option is A $c o s^{- 1} (\frac{1}{\sqrt{3}})$
It is given that, $| \to a | = | \to b | = | \to c | = λ$ (say) and $\to a, \to b, \to c$ are mutually perpendicular vectors.
Therefore, $| \to a + \to b + \to c | = \sqrt{3} λ$
Let $θ$ be the angle which $\to a + \to b + \to c$ makes with $\to a$ .
Then,
$c o s θ = \frac{\to a . (\to a + \to b + \to c)}{| \to a | | \to a + \to b + \to c |} = \frac{| \to a |^{2}}{| \to a | | \to a + \to b + \to c |} = \frac{λ^{2}}{λ (\sqrt{3} λ)} = \frac{1}{\sqrt{3}} ∴ θ = c o s^{- 1} (\frac{1}{\sqrt{3}})$

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If →a,→b,→c are three mutually perpendicular vectors of equal magnitude, then the angle θ which →a+→b+→c makes with any one of three given vectors is given by

If $\to a, \to b, \to c$ are three mutually perpendicular vectors of equal magnitude, then the angle $θ$ which $\to a + \to b + \to c$ makes with any one of three given vectors is given by