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 $\vec{a} = 2\hat{i} - 3\hat{j} + \hat{k}$ and $\vec{b} = 2\hat{i} + \hat{j} - \hat{k}$, then which of the following statements is/are correct?

(A) $\vec{a}$ and $\vec{b}$ are collinear

(B) $\vec{a}$ and $\vec{b}$ are perpendicular

(C) Angle between $\vec{a}$ and $\vec{b}$ is $\frac{\pi}{4}$

(D) $|\vec{a} + \vec{b}| = 2\sqrt{5}$

Choose the correct answer from the options given below:

Match List-I with List-II

Choose the correct answer from the options given below:

For any vector $\vec{a}$, the value of $|\vec{a} \times \hat{i}|^2 + |\vec{a} \times \hat{j}|^2 + |\vec{a} \times \hat{k}|^2$ is equal to:

The position vector of a point R which divides the line joining two points P and Q whose position vectors are $\hat{i} + 2\hat{j} - \hat{k}$ and $-\hat{i} + \hat{j} + \hat{k}$ respectively in the ratio 2 : 1 externally is :

Let $\vec{a} = 2\hat{i} + 3\hat{j} + \hat{k}$ and $\vec{b} = 2\hat{i} - \hat{j} + 2\hat{k}$ be two vectors. Then which of the following are TRUE?

A. $\vec{a}$ and $\vec{b}$ are perpendicular

B. Projection of $\vec{a}$ on $\vec{b}$ is 1

C. $|\vec{a} \times \vec{b}| = 3\sqrt{13}$

D. $\vec{a} - \vec{b}$ is a unit vector

Choose the correct answer from the options given below:

If $\vec{a} = \hat{i} - 2\hat{j} + 3\hat{k}$ and $\vec{b} = \hat{i} + 2\hat{j} + \hat{k}$, then which of the following statements are correct?

(A) $\vec{a}$ and $\vec{b}$ are perpendicular to each other.

(B) $\vec{a}$ and $\vec{b}$ are parallel.

(C) $|\vec{a}\times\vec{b}| = 2\sqrt{21}$.

(D) Projection of $\vec{a}$ on $\vec{b}$ = 0.

Choose the correct answer from the options given below: