Let a→=i^+j^;b→=2i^-k^. Then vector r→satisffing the equations r→×a→=b→×a→ and r→×b→=a→×b→ is

A
$\hat{i} - \hat{j} + \hat{k}$
B
$3 \hat{i} - \hat{j} + \hat{k}$
C
$3 \hat{i} + \hat{j} - \hat{k}$
D
$\hat{i} - \hat{j} - \hat{k}$

Solution:

$\vec{r} \times \vec{a} = \vec{b} \times \vec{a} o r (\vec{r} - \vec{b}) \times \vec{a} = 0$

$\vec{r} \times \vec{b} = \vec{a} \times \vec{b} o r (\vec{r} - \vec{a}) \times \vec{b} = 0$

If $\vec{r} = x \hat{i} + y \hat{j} + z \hat{k}$ , then

$|\begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ x - 2 & y & z + 1 \\ 1 & 1 & 0 \end{matrix}| = 0 a n d |\begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ x - 1 & y - 1 & z \\ 2 & 0 & - 1 \end{matrix}| = 0$

$\begin{array}{l} \Rightarrow z + 1 = 0, x - y = 2 \\ a n d y - 1 = 0, x - 1 + 2 z = 0 \\ \Rightarrow x = 3, y = 1, z = - 1 \end{array}$

Let $\vec{a} = \hat{i} + \hat{j}; \vec{b} = 2 \hat{i} - \hat{k}$ . Then vector $\vec{r}$ satisffing the equations $\vec{r} \times \vec{a} = \vec{b} \times \vec{a}$ and $\vec{r} \times \vec{b} = \vec{a} \times \vec{b}$ is

Solution:

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