If $\vec{a} = 2\hat{i} - \hat{j} + 3\hat{k}$ and $\vec{b} = 2\hat{i} + 2\hat{j} + \hat{k}$, then Match List-I with List-II | List-I | List-II | |---|---| | (A) Projection of $\vec{a}$ on $\vec{b}$ is | (I) $-7\hat{i} + 4\hat{j} + 6\hat{k}$ | | (B) $\vec{a} \times \vec{b}$ is | (II) $\frac{1}{\sqrt{101}}(-7\hat{i} + 4\hat{j} + 6\hat{k})$ | | (C) unit vector along $\vec{a} + \vec{b}$ is | (III) $\frac{5}{3}$ | | (D) Unit vector perpendicular to both $\vec{a}$ & $\vec{b}$ is | (IV) $\frac{1}{\sqrt{33}}(4\hat{i} + \hat{j} + 4\hat{k})$ | Choose the correct answer from the options given below:

Let $\vec{a}$ and $\vec{b}$ be two unit vectors. If the vectors $\vec{c} = 5\vec{a} - 4\vec{b}$ and $\vec{d} = \vec{a} + 2\vec{b}$ are perpendicular to each other, then the angle between $\vec{a}$ and $\vec{b}$ is :

The position vectors of the points A and B are $\vec{a}=2\hat{i}-3\hat{j}+2\hat{k}$ and $\vec{b}=2\hat{i}+3\hat{j}+\hat{k}$ respectively, then area of triangle OAB 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

Let $\vec{a}, \vec{b}$ be two vectors such that $|\vec{a}| = 2, |\vec{b}| = 3, \vec{a} \cdot \vec{b} = 4$, then $|\vec{a} - \vec{b}|$ is equal to

If $\vec{a}, \vec{b}, \vec{c}$ are unit vectors such that $\vec{a} + \vec{b} + \vec{c} = 0$. The value of $\vec{a}.\vec{b} + \vec{b}.\vec{c} + \vec{c}.\vec{a}$ is

A unit vector perpendicular to the vectors $\hat{i} - \hat{j}$ and $\hat{i} + \hat{j}$ is