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Direction Cosines

If a vector ...

Question

If a vector $\to a$ is expressed as the sum of two vectors $\to α$ and $\to β$ along and perpendicular to a given vector $\to b$ , then $\to β =$

A

\frac{(\to a \times \to b) \times \to b}{| \to b |^{2}}

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B

\frac{(\to b \times \to a) \times \to b}{| \to b |^{2}}

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C

\frac{(\to b \times \to a) \times \to b}{| \to b |}

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D

\frac{(\to a . \to b) \to b}{| \to b |^{2}}

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Solution

The correct option is B $\frac{(\to b \times \to a) \times \to b}{| \to b |^{2}}$
The perpendicular component will be
$= β = ¯ ¯ ¯ a - \frac{(¯ ¯ ¯ a . ¯ ¯ b) ¯ ¯ b}{b^{2}}$
$= \frac{¯ ¯ ¯ a (¯ ¯ b . ¯ ¯ b) - (¯ ¯ ¯ a . ¯ ¯ b) ¯ ¯ b}{b^{2}}$
$= \frac{¯ ¯ b \times (¯ ¯ ¯ a \times ¯ ¯ b)}{b^{2}}$

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