If $|\vec{a}| = |\vec{b}| = |\vec{a} + \vec{b}| = 1$, then $|\vec{a} - \vec{b}|$ 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:

If $\vec a, \vec b, \vec c$ are mutually perpendicular vectors of equal magnitude, the angle between $\vec a+\vec b+\vec c$ and $\vec a$ is

If $\hat{i}$, $\hat{j}$ and $\hat{k}$ are unit vectors along the co-ordinate axes OX, OY and OZ respectively, then

(A) $\hat{i} \times \hat{j} = \hat{k}$

(B) $\hat{k} \times \hat{i} = -\hat{j}$

(C) $\hat{j} \cdot \hat{j} = 1$

(D) $\hat{j} \cdot \hat{k} = 0$

Choose the correct answer from the options given below:

The area (in sq.units) of a triangle formed by vertices O, A and B where $\vec{OA} = \hat{i} + 2\hat{j} + 3\hat{k}$ and $\vec{OB} = -3\hat{i} - 2\hat{j} + \hat{k}$ is

If $|\vec{a}| = 10$, $|\vec{b}| = 2$ and $\vec{a} \cdot \vec{b} = 12$, then value of $|\vec{a} \times \vec{b}|$ is :

If a unit vector $\hat a$ makes an angle $\dfrac{\pi}{3}$ with $\hat i$, $\dfrac{\pi}{4}$ with $\hat j$ and an acute angle $\theta$ with $\hat k$, then the respective value of $\theta$ and $\hat a$ are