Question

Given the following simultaneous equations for vectors x and y

$\mathbf{x + y = a}$ .....(i)

$\mathbf{x}\mathbf{\times}\mathbf{y = b}$ .....(ii)

$\mathbf{x.a = 1}$ .....(iii)

Then $\mathbf{x} = .........,\mathbf{y} = .......$

• A. $\mathbf{a},\mathbf{a} - \mathbf{x}$
• B. $\mathbf{a} - \mathbf{b},\mathbf{b}$
• C. $\mathbf{b},\mathbf{a} - b$
• D. None of these

Answer 1

Answer: (D)

None of these

Answer 2

Multiplying (i) scalarly by $\mathbf{a},$ we get $\mathbf{a}.\mathbf{x} + \mathbf{a}.\mathbf{y} = \mathbf{a}^{2}$

$\therefore\mathbf{a}.\mathbf{y} = \mathbf{a}^{2} - 1$ …..(iv), {By (iii)}

Again $\mathbf{a} \times (\mathbf{x} \times \mathbf{y}) = \mathbf{a} \times \mathbf{b}$or $(\mathbf{a}.\mathbf{y})\mathbf{x} - (\mathbf{a}.\mathbf{x})\mathbf{y} = \mathbf{a} \times \mathbf{b}$

$(\mathbf{a}^{2} - 1)\mathbf{x} - \mathbf{y} = \mathbf{a} \times \mathbf{b}$ …..(v), {By (iii) and (iv)}

Adding and subtracting (i) and (v), we get $x = \frac{\mathbf{a} + (\mathbf{a} \times \mathbf{b})}{\mathbf{a}^{2}}$ and $\mathbf{y} = \mathbf{a} - \mathbf{x}$ etc.