Question

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

• A. $l = \frac{(\mathbf{c} \times \mathbf{b}).(\mathbf{a} \times \mathbf{b})}{(\mathbf{a} \times \mathbf{b})^{2}},m = \frac{(\mathbf{c} \times \mathbf{a}).(\mathbf{b} \times \mathbf{a})}{(\mathbf{b} \times \mathbf{a})^{2}}$
• B. $l = \frac{(\mathbf{c} \times \mathbf{b}).(\mathbf{a} \times \mathbf{b})}{(\mathbf{a} \times \mathbf{b})},m = \frac{(\mathbf{c} \times \mathbf{a}).(\mathbf{b} \times \mathbf{a})}{(\mathbf{b} \times \mathbf{a})}$
• C. $l = \frac{(\mathbf{c} \times \mathbf{b}) \times (\mathbf{a} \times \mathbf{b})}{(\mathbf{a} \times \mathbf{b})^{2}},m = \frac{(\mathbf{c} \times \mathbf{a}) \times (\mathbf{b} \times \mathbf{a})}{(\mathbf{b} \times \mathbf{a})}$
• D. None of these

Answer 1

Answer: (A)

$l = \frac{(\mathbf{c} \times \mathbf{b}).(\mathbf{a} \times \mathbf{b})}{(\mathbf{a} \times \mathbf{b})^{2}},m = \frac{(\mathbf{c} \times \mathbf{a}).(\mathbf{b} \times \mathbf{a})}{(\mathbf{b} \times \mathbf{a})^{2}}$

Answer 2

Here $(l\mathbf{a} + m\mathbf{b}) \times \mathbf{b} = \mathbf{c} \times \mathbf{b} \Rightarrow l\mathbf{a} \times \mathbf{b} = \mathbf{c} \times \mathbf{b}$

$\Rightarrow l(\mathbf{a} \times \mathbf{b})^{2} = (\mathbf{c} \times \mathbf{b}).(\mathbf{a} \times \mathbf{b}) \Rightarrow l = \frac{(\mathbf{c} \times \mathbf{b}).(\mathbf{a} \times \mathbf{b})}{(\mathbf{a} \times \mathbf{b})^{2}}$

Similarly, $m = \frac{(\mathbf{c} \times \mathbf{a}).(\mathbf{b} \times \mathbf{a})}{(\mathbf{b} \times \mathbf{a})^{2}}$.