Question

If $a = i + j$ and $b = 2i - k$ are two vectors, then the point of intersection of two lines $r \times a = b \times a$ and $r \times b = a \times b$ is

• A. i + j – k
• B. i – j + k
• C. 3i + j – k
• D. 3i – j + k

Answer 1

Answer: (C)

3i + j – k

Answer 2

Let $\mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}$, then $\mathbf{r} \times \mathbf{a} = \mathbf{b} \times \mathbf{a}$

⇒ $(\mathbf{r} - \mathbf{b}) \times \mathbf{a} = 0$ $\Rightarrow \left| \begin{aligned} & ijk \\ & x - 2yz + 1 \\ & 110 \end{aligned} \right| = 0$,

$\therefore z = - 1,x - y = 2$

Now $\mathbf{r} \times \mathbf{b} = \mathbf{a} \times \mathbf{b}$ ⇒ $(\mathbf{r} - \mathbf{a}) \times \mathbf{b} = 0$

$\therefore y = 1,x + 2z = 1$ $\Rightarrow x = 3,y = 1,z = - 1$

∴$\mathbf{r} = 3\mathbf{i} + \mathbf{j} - \mathbf{k}$.