Let a- b and c be three unit vectors- out of which vectors b and c are non-parallel- If - and - are the angles which vector a makes with vectors b and c respectively and a- b-c -12b- then - - - is equal to -

A× ( b×c )=12b ⇒ (a·c nbsp; )b (a·b nbsp; )c=12b ⇒ (a·c 12 nbsp; )b (a·b nbsp; )c=0 As vectors b and c are non parallel, ∴a·c 12 nbsp;=0 ⇒a·c=12⇒co ...

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Byju's Answer

Standard XIII

Mathematics

Linear Dependence and Independence of Vectors

Let a, b and ...

Question

Let $\to a, \to b$ and $\to c$ be three unit vectors, out of which vectors $\to b$ and $\to c$ are non-parallel. If $α$ and $β$ are the angles which vector $\to a$ makes with vectors $\to b$ and $\to c$ respectively and $\to a \times (\to b \times \to c) = \frac{1}{2} \to b$ , then $| α - β |$ is equal to :

60^{\circ}

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45^{\circ}

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30^{\circ}

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90^{\circ}

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Solution

The correct option is C $30^{\circ}$
$\to a \times (\to b \times \to c) = \frac{1}{2} \to b$
$\Rightarrow (\to a \cdot \to c) \to b - (\to a \cdot \to b) \to c = \frac{1}{2} \to b$
$\Rightarrow (\to a \cdot \to c - \frac{1}{2}) \to b - (\to a \cdot \to b) \to c = 0$

As vectors $\to b$ and $\to c$ are non-parallel,
$∴ \to a \cdot \to c - \frac{1}{2} = 0$
$\Rightarrow \to a \cdot \to c = \frac{1}{2} \Rightarrow cos β = \frac{1}{2}$
$∴ β = 60^{\circ}$
and $\to a \cdot \to b = 0 \Rightarrow α = 90^{\circ}$
$| α - β | = 30^{\circ}$

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Let →a, →b and →c be three unit vectors, out of which vectors →b and →c are non-parallel. If α and β are the angles which vector →a makes with vectors →b and →c respectively and →a×(→b×→c)=12→b, then |α−β| is equal to :