Let → a , → b , → c be three vectors mutually perpendicular to each...

Let

\overset{\to}{a}, \overset{\to}{b}, \overset{\to}{c}

be three vectors mutually perpendicular to each other and have same magnitude. If a vector

\overset{\to}{r}

satisfies

\overset{\to}{a} \times { \left(\overset{\to}{r} - \overset{\to}{b}\right) \times \overset{\to}{a} } + \overset{\to}{b} \times { \left(\overset{\to}{r} - \overset{\to}{c}\right) \times \overset{\to}{b} } + \overset{\to}{c} \times { \left(\overset{\to}{r} - \overset{\to}{a}\right) \times \overset{\to}{c} } = \overset{\to}{0},

then

\overset{\to}{r}

is equal to:

Options

A $\dfrac{1}{3} ( \overset{\to}{a} + \overset{\to}{b} + \overset{\to}{c} )$
B $\dfrac{1}{3} ( 2\overset{\to}{a} + \overset{\to}{b} - \overset{\to}{c} )$
C $\dfrac{1}{2} ( \overset{\to}{a} + \overset{\to}{b} + \overset{\to}{c} )$
D $\dfrac{1}{2} ( \overset{\to}{a} + \overset{\to}{b} + 2\overset{\to}{c} )$

Correct answer

C. $\dfrac{1}{2} ( \overset{\to}{a} + \overset{\to}{b} + \overset{\to}{c} )$

Step-by-step solution

Given: $\overset{\to}{a} \times { \left(\overset{\to}{r} - \overset{\to}{b}\right) \times \overset{\to}{a} } + \overset{\to}{b} \times { \left(\overset{\to}{r} - \overset{\to}{c}\right) \times \overset{\to}{b} } + \overset{\to}{c} \times { \left(\overset{\to}{r} - \overset{\to}{a}\right) \times \overset{\to}{c} } = \overset{\to}{0}$

$⇒ ( \overset{\to}{a} . \overset{\to}{a} ) ( \overset{\to}{r} - \overset{\to}{b} ) - ( \overset{\to}{a} \cdot ( \overset{\to}{r} - \overset{\to}{b} ) ) \overset{\to}{a} + ( \overset{\to}{b} \cdot \overset{\to}{b} ) ( \overset{\to}{r} - \overset{\to}{c} ) - ( \overset{\to}{b} \cdot ( \overset{\to}{r} - \overset{\to}{c} ) ) \overset{\to}{b} + ( \overset{\to}{c} . \overset{\to}{c} ) ( \overset{\to}{r} - \overset{\to}{a} ) - ( \overset{\to}{c} . ( \overset{\to}{r} - \overset{\to}{a} ) ) \overset{\to}{c} = \overset{\to}{0}$

$⇒ | \overset{\to}{a}{|}^{2} ( \overset{\to}{r} - \overset{\to}{b} ) - ( \overset{\to}{r} \cdot \overset{\to}{a} ) \overset{\to}{a} + | \overset{\to}{b}{|}^{2} ( \overset{\to}{r} - \overset{\to}{c} ) - ( \overset{\to}{r} \cdot \overset{\to}{b} ) \overset{\to}{b} + | \overset{\to}{c}{|}^{2} ( \overset{\to}{r} - \overset{\to}{a} ) - ( \overset{\to}{r} \cdot \overset{\to}{c} ) \overset{\to}{c} = \overset{\to}{0}$

$⇒ | \overset{\to}{a}{|}^{2} [ 3\overset{\to}{r} - ( \overset{\to}{a} + \overset{\to}{b} + \overset{\to}{c} ) ] - [ ( \overset{\to}{r} \cdot \overset{\to}{a} ) \overset{\to}{a} ) + ( \overset{\to}{r} \cdot \overset{\to}{b} ) \overset{\to}{b} + ( \overset{\to}{r} \cdot \overset{\to}{c} ) \overset{\to}{c} ] = \overset{\to}{0}\left[∵ | \overset{\to}{a}{|}^{2} = | \overset{\to}{b}{|}^{2} = | \overset{\to}{c}{|}^{2}\right]$

$⇒ | \overset{\to}{a}{|}^{2} [ 3\overset{\to}{r} - ( \overset{\to}{a} + \overset{\to}{b} + \overset{\to}{c} ) - ( \overset{\to}{r} ) ] = \overset{\to}{0}\left[∵ ( \overset{\to}{r} \cdot \overset{\to}{a} ) \overset{\to}{a} + ( \overset{\to}{r} \cdot \overset{\to}{b} ) \overset{\to}{b} + ( \overset{\to}{r} \cdot \overset{\to}{c} ) \overset{\to}{c} = {\left|\overset{\to}{a}\right|}^{2}\overset{\to}{r}, | \overset{\to}{a}{|}^{2} = | \overset{\to}{b}{|}^{2} = | \overset{\to}{c}{|}^{2}\right]$

$⇒ 3\overset{\to}{r} - ( \overset{\to}{a} + \overset{\to}{b} + \overset{\to}{c} ) - \overset{\to}{r} = \overset{\to}{0}$

$∴ \overset{\to}{r} = \dfrac{\overset{\to}{a} + \overset{\to}{b} + \overset{\to}{c}}{2}$

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