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Definition of Vector

If a⃗, b⃗, ...

Question

If $\to a, \to b, \to c$ are three mutually perpendicular vectors of equal magnitude, prove that $(\to a + \to b + \to c)$ is equally inclined with vectors $\to a, \to b$ and $\to c$ .

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Solution

$\to a . \to b = \to b . \to c = \to c . \to a = 0$

Let the angle between $\to a$ and $(\to a + \to b + \to c)$ be $α$

$\to a . (\to a + \to b + \to c) = | \to a | | \to a + \to b + \to c | cos α \to a . \to a + \to a . \to b + \to c . \to a = | \to a | | \to a + \to b + \to c | cos α {| \to a |}^{2} + 0 + 0 = | \to a | | \to a + \to b + \to c | cos α \Rightarrow cos α = \frac{| \to a |}{| \to a + \to b + \to c |}$

Similarly angle between $\to b$ and $(\to a + \to b + \to c)$ be $β$

$\Rightarrow cos β = \frac{| \to b |}{| \to a + \to b + \to c |}$

And angle between $\to c$ and $(\to a + \to b + \to c)$ be $γ$

$\Rightarrow cos γ = \frac{| \to c |}{| \to a + \to b + \to c |}$

As all the vectors are equal in magnitude

$\Rightarrow cos α = cos β = cos γ \Rightarrow α = β = γ$

Hence proved.

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