Let a- b- c- be three vectors having magnitude 1- 1 and 2 respectively- If a- a-c- -b-0- the acute angle between a- and c- is

The correct option is C π6 From the information given, we have #160; b⃗ = #160;a⃗× (a⃗ #160;× #160;c⃗) ⇒ b⃗ = (a⃗.c⃗) #160;a⃗ (a⃗ ...

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Byju's Answer

Standard XII

Mathematics

Angle between Two Line Segments

Let a⃗, b⃗,...

Question

Let $\to a, \to b, \to c$ be three vectors having magnitude $1, 1$ and $2$ respectively. If $\to a \times (\to a \times \to c) + \to b = \to 0$ , the acute angle between $\to a$ and $\to c$ is

\frac{π}{3}

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\frac{π}{4}

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\frac{π}{6}

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None of these

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Solution

The correct option is C $\frac{π}{6}$
From the information given, we have $- \to b = \to a \times (\to a \times \to c)$

$\Rightarrow - \to b = (\to a . \to c) \to a - (\to a . \to a) \to c$
$\Rightarrow | - \to b | = | (\to a . \to c) \to a - (\to a . \to a) \to c |$
$\Rightarrow b^{2} = (\to a . \to c)^{2} a^{2} + (\to a . \to a)^{2} c^{2} - 2 a^{2} (\to a . \to c)^{2}$
$\Rightarrow 1 = 4 {cos}^{2} θ + 4 - 8 {cos}^{2} θ$
$\Rightarrow {cos}^{2} θ = \frac{3}{4}$
$\Rightarrow θ = \frac{π}{6}$

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Similar questions

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Let →a,→b,→c be three vectors having magnitude 1,1 and 2 respectively. If →a×(→a×→c)+→b=→0, the acute angle between →a and →c is

The correct option is C π6From the information given, we have −→b=→a×(→a×→c)⇒−→b=(→a.→c)→a−(→a.→a)→c⇒|−→b|=|(→a.→c)→a−(→a.→a)→c|⇒b2=(→a.→c)2a2+(→a.→a)2c2−2a2(→a.→c)2⇒1=4cos2θ+4−8cos2θ⇒cos2θ=34⇒θ=π6

Let $\to a, \to b, \to c$ be three vectors having magnitude $1, 1$ and $2$ respectively. If $\to a \times (\to a \times \to c) + \to b = \to 0$ , the acute angle between $\to a$ and $\to c$ is