A- b- c are three unit vectors such that a- b- c - 1 - 2 b-c - If the vectors b and c are non-parallel- then the angle between a and b is equal to

The correct option is B #160;2π #160;/ 3 Let angles between a and b and between a and c br α and β respectively.We h ...

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Byju's Answer

Standard XII

Mathematics

Definition of Functions

a, b, c are t...

Question

$a, b, c$ are three unit vectors such that $a \times (b \times c) = \frac{1}{2} (b + c)$ . If the vectors $b$ and $c$ are non-parallel, then the angle between $a$ and $b$ is equal to

\frac{π}{3}

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\frac{2 π}{3}

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\frac{5 π}{6}

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none of these

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Solution

The correct option is B $\frac{2 π}{3}$
Let angles between $a$ and $b$ and between $a$ and $c$ br $α$ and $β$ respectively.
We have, $a \times (b \times c) = \frac{1}{2} (b + c)$
$\Rightarrow (a . c) b - (a . b) c = \frac{1}{2} b + \frac{1}{2} c$
$\Rightarrow (a . c - \frac{1}{2}) b - (a . b - \frac{1}{2}) c = 0$
$\Rightarrow a . c = \frac{1}{2}, a . b = - \frac{1}{2}$ $($ As $b$ and $c$ are non parallel $)$
$\Rightarrow cos β = \frac{1}{2}, cos α = - \frac{1}{2} [a, b, c$ are unit vectors $]$
$\Rightarrow β = \frac{π}{3}, α = \frac{2 π}{3}$
Hence angle between $a$ and $b$ is $\frac{2 π}{3}$

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$a, b, c$ are three unit vectors such that $a \times (b \times c) = \frac{1}{2} (b + c)$ . If the vectors $b$ and $c$ are non-parallel, then the angle between $a$ and $b$ is equal to