Let $\vec{a} = \hat{i} + 4\hat{j} + 2\hat{k}$, $\vec{b} = 3\hat{i} - 2\hat{j} + 7\hat{k}$ and $\vec{c} = 2\hat{i} + \hat{j} + 4\hat{k}$. A vector $\vec{d}$ which is perpendicular to both $\vec{a}$ and $\vec{b}$, and $\vec{c} \cdot \vec{d} = 14$, 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:

Match List-I with List-II

Choose the correct answer from the options given below:

If $\vec{a}, \vec{b}, \vec{c}$ are three mutually perpendicular unit vectors, then $|\vec{a} + \vec{b} + \vec{c}|$ is equal to

The value of $\hat{i} \cdot (\hat{k} \times \hat{j}) + \hat{j} \cdot (\hat{i} \times \hat{k}) + \hat{k} \cdot (\hat{j} \times \hat{i})$ is

Let $\vec{OA} = 2\hat{i} - \hat{j} + \hat{k}$ and $\vec{OB} = \hat{i} + \hat{j} - \hat{k}$. Then

A. The magnitude of vector $\vec{OA}$ is 6

B. The magnitude of vector $\vec{OB}$ is $\sqrt{3}$

C. The vector $\vec{AB}$ is $(-\hat{i} + 2\hat{j} - 2\hat{k})$

D. $\vec{OA} \cdot \vec{OB} = 0$

E. $\vec{OA} \parallel \vec{OB}$

Choose the correct answer from the options given below:

If $\vec{a}$ and $\vec{b}$ are two collinear vectors, then which of the following statements is/are incorrect?

(A) $\vec{b} = \lambda\vec{a}$, for some scalar $\lambda$.

(B) Always $\vec{a} = \pm\vec{b}$.

(C) the respective components of $\vec{a}$ and $\vec{b}$ are proportional.

(D) both vectors $\vec{a}$ and $\vec{b}$ always have same direction.

Choose the correct answer from the options given below: