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Applications of Cross Product

If non-zero v...

Question

If non-zero vectors $\to a$ and $\to b$ are perpendicular to each other, then the solution of the equation $\to r \times \to a = \to b$ is given by

\to r = x \to a + \frac{\to a \times \to b}{| \to a |^{2}}

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\to r = x \to b + \frac{\to a \times \to b}{| \to b |^{2}}

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\to r = x (\to a \times \to b)

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\to r = x (\to b \times \to a)

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Solution

The correct option is A $\to r = x \to a + \frac{\to a \times \to b}{| \to a |^{2}}$
As $\to a ⊥ \to b$
$∴ \to a \cdot \to b = 0 . . . . ()$
Again $\to a, \to b, \to a \times \to b$ are non coplanar
$∴ \to r = x \to a + y \to b + z (\to a \times \to b) . . . . (1)$
for some $x, y, z ϵ R$
$∴ \to b = \to r \times \to a$
$\Rightarrow \to b = (x \to a + y \to b + z (\to a \times \to b)) \times \to a$
$= y (\to b \times \to a) + z ((\to a \times \to b) \times \to a)$
$= y (\to b \times \to a) - z (\to a \times (\to a \times \to b))$
$= y (\to b \times \to a) - z [(\to a \cdot \to b) \to a - (\to a \cdot \to a) \to b]$
$\Rightarrow \to b = y (\to b \times \to a) + z (\to a \cdot \to a) \to b$ ....... using $()$
$\Rightarrow y = 0, z = \frac{1}{\to a \cdot \to a} = \frac{1}{| \to a |^{2}}$
$∴$ From $(1)$ we get $\to r = x \to a + \frac{1}{| \to a |^{2}} (\to a \times \to b)$
Hence, option A is correct.

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