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Cross Product of Two Vectors

Let a⃗, b⃗ an...

Question

Let $\to a, \to b$ and $\to c$ be three vectors such that $| \to a | = \sqrt{3}, | \to b | = 5, \to b . \to c = 10$ and the angle between $\to b$ and $\to c$ is $\frac{π}{3} .$
If $\to a$ is perpendicular to vector $\to b \times \to c$ , then $| \to a \times (\to b \times \to c) |$ is equal to

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Solution

$| \to a \times (\to b \times \to c) | = | \to a | | \to b \times \to c | sin θ$ where $θ$ is angle between $\to a and \to b \times \to c$
$θ = \frac{π}{2}$ (given)
$\Rightarrow | \to a \times (\to b \times \to c) | = \sqrt{3} | \to b \times \to c | sin \frac{π}{3}$
$\Rightarrow | \to a \times (\to b \times \to c) | = \sqrt{3} \times 5 \times | \to c | \times \frac{\sqrt{3}}{2}$
$\Rightarrow | \to a \times (\to b \times \to c) | = \frac{15}{2} | \to c |$
Now, $| \to b | | \to c | cos θ = 10$
$\Rightarrow 5 | \to c | \frac{1}{2} = 10$
$\Rightarrow | \to c | = 4$
$\Rightarrow | \to a \times (\to b \times \to c) | = 30$

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Cross Product of Two Vectors

Standard XII Mathematics

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