Let $\vec{a}=\hat{i}-\hat{j}$, $\vec{b}=3\hat{j}-\hat{k}$ and $\vec{c}=7\hat{i}-\hat{k}$. Then a vector $\vec{d}$ which is perpendicular to both $\vec{a}$ and $\vec{b}$ and $\vec{c}.\vec{d}=1$ is

If $\vec{a}$ is a unit vector perpendicular to both the vectors $\vec{b} = \hat{j} + \hat{2k}$ and $\vec{c} = \hat{i} + 2\hat{j}$, then $\hat{a}$ is equal to

If $\vec{a} = \hat{i} - 7\hat{j} + 4\hat{k}$ and $\vec{b} = 2\hat{i} - 14\hat{j} - p\hat{k}$ are parallel, then the value of $p$, is:

The vector equation of the line passing through points $A(3,4,-7)$ and $B(1,-1,6)$ is

A unit vector perpendicular to both the vectors $4\hat{i}-\hat{j}+3\hat{k}$ and $-2\hat{i}+\hat{j}-2\hat{k}$ is:

The position vector of the point which divides the join of the points $2\vec{a}-3\vec{b}$ and $3\vec{a}-2\vec{b}$ externally in the ratio 2:3, is:

Let $\vec{a} = 4\hat{i} - \hat{j} + 3\hat{k}$ and $\vec{b} = -2\hat{i} + \hat{j} - 2\hat{k}$. Then

(A) $\vec{a}$ is a unit vector

(B) $\vec{a} \times \vec{b} = -\hat{i} + 2\hat{j} + 2\hat{k}$

(C) $\vec{a}$ and $\vec{b}$ are parallel vectors

(D) $\vec{a}$ and $\vec{b}$ are neither parallel nor perpendicular vectors

Choose the correct answer from the options given below :