Let a- b- and c- be three non-zero vectors such that no two of them are collinear and a-b- -c- - 1-3- b- c-a-If - is the angle between vectors b- and c- then a value of sin- is

The correct option is A 2√(2)/3 (a⃗×b⃗)×c⃗=13|b⃗||c⃗|a⃗ ⇒ a⃗(b⃗.c⃗)+b⃗(a⃗.c⃗)=13|b⃗||c⃗|a⃗ Comparing both sides, we get (a⃗·c⃗)=0 #160; ...

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Byju's Answer

Standard XII

Mathematics

Integration by Parts

Let a⃗, b⃗ ...

Question

Let $\to a, \to b$ and $\to c$ be three non-zero vectors such that no two of them are collinear and $(\to a \times \to b) \times \to c = \frac{1}{3} ∣ ∣ \to b ∣ ∣ | \to c | \to a$ .
If $θ$ is the angle between vectors $\to b$ and $\to c$ , then a value of $sin θ$ is

\frac{2 \sqrt{2}}{3}

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\frac{\sqrt{2}}{3}

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\frac{2}{3}

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\frac{- 2 \sqrt{3}}{3}

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Solution

The correct option is A $\frac{2 \sqrt{2}}{3}$
$(\to a \times \to b) \times \to c = \frac{1}{3} ∣ ∣ \to b ∣ ∣ | \to c | \to a$
$\Rightarrow - \to a (\to b . \to c) + \to b (\to a . \to c) = \frac{1}{3} ∣ ∣ \to b ∣ ∣ | \to c | \to a$

Comparing both sides, we get
$(\to a \cdot \to c) = 0$ and $- \to a ∣ ∣ \to b ∣ ∣ | \to c | cos θ = \frac{1}{3} ∣ ∣ \to b ∣ ∣ | \to c | \to a$

$- \to a ∣ ∣ \to b ∣ ∣ | \to c | cos θ = \frac{1}{3} ∣ ∣ \to b ∣ ∣ | \to c | \to a$ $\Rightarrow cos θ = - \frac{1}{3}$
$sin θ = \sqrt{1 - {cos}^{2} θ} = \sqrt{1 - {(\frac{1}{3})}^{2}} = \sqrt{\frac{9 - 1}{9}} = \frac{\sqrt{8}}{3} = \frac{2 \sqrt{2}}{3}$

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Let →a,→b and →c be three non-zero vectors such that no two of them are collinear and (→a×→b)×→c=13∣∣→b∣∣|→c|→a.If θ is the angle between vectors →b and →c, then a value of sinθ is

Let $\to a, \to b$ and $\to c$ be three non-zero vectors such that no two of them are collinear and $(\to a \times \to b) \times \to c = \frac{1}{3} ∣ ∣ \to b ∣ ∣ | \to c | \to a$ .
If $θ$ is the angle between vectors $\to b$ and $\to c$ , then a value of $sin θ$ is