The scalars l and m such that l a-mb-c- where a- b- and c- are given vectors- are equal to

La⃗+mb⃗ =c⃗ (la⃗+mb⃗)×b⃗=c⃗×b⃗⇒ l a⃗×b⃗=c⃗×b⃗ Take dot product with a⃗×b⃗ ⇒ l|a⃗×b⃗|^2=(c⃗×b⃗).(a⃗×b⃗) ⇒ l=(c⃗×b⃗).(a⃗×b⃗)/|a⃗×b⃗|^2 Similarly, m= (c⃗×a⃗).(b⃗×a ...

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Standard XI

Mathematics

Cross Product of Two Vectors

The scalars l...

Question

The scalars l and m such that $l \to a + m \to b = \to c$ where $\to a, \to b$ and $\to c$ are given vectors, are equal to

$l = \frac{(\to c \times \to b) . (\to a \times \to b)}{| \to a \times \to b |^{2}}$ ; $m = \frac{(\to c \times \to a) . (\to b \times \to a)}{| \to b \times \to a |^{2}}$

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$l = \frac{(\to c \times \to b) . (\to a \times \to b)}{| \to a \times \to b |}$ ; $m = \frac{(\to c \times \to a) . (\to b \times \to a)}{| \to b \times \to a |}$

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$l = \frac{| \to c \times \to b |^{2}}{| \to a \times \to b |^{2}}$ ; $m = \frac{| \to c \times \to a |^{2}}{| \to b \times \to a |^{2}}$

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$l = \frac{| \to c \times \to a |^{2}}{| \to b \times \to a |^{2}}$ ; $m = \frac{| \to c \times \to b |^{2}}{| \to a \times \to b |^{2}}$

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Solution

The correct option is A
$l = \frac{(\to c \times \to b) . (\to a \times \to b)}{| \to a \times \to b |^{2}}$ ; $m = \frac{(\to c \times \to a) . (\to b \times \to a)}{| \to b \times \to a |^{2}}$

$l \to a + m \to b = \to c$
$(l \to a + m \to b) \times \to b = \to c \times \to b \Rightarrow l \to a \times \to b = \to c \times \to b$
Take dot product with $\to a \times \to b$
$\Rightarrow l | \to a \times \to b |^{2} = (\to c \times \to b) . (\to a \times \to b) \Rightarrow l = \frac{(\to c \times \to b) . (\to a \times \to b)}{| \to a \times \to b |^{2}}$
Similarly, $m = \frac{(\to c \times \to a) . (\to b \times \to a)}{| \to b \times \to a |^{2}}$

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