Match List-I with List-II | List-I | List-II | | --- | --- | | (A) If vector $\vec{a}$ and $\vec{b}$ are such that $\vec{a} = \lambda \vec{b}$ and $\vert \vec{a}\vert = \vert \vec{b}\vert $, then | (I) $\vec{a}$ and $\vec{b}$ are orthogonal | | (B) Projection vector of $\vec{a}$ on $\vec{b}$ | (II) $[0, 12]$ | | (C) $\vec{a}$ and $\vec{b}$ are non-zero vectors such that $\vert \vec{a} + \vec{b}\vert = \vert \vec{a} - \vec{b}\vert $, then | (III) $\vec{a} = \pm \vec{b}$ | | (D) If $\vert \vec{a}\vert = 4, -3 \le \lambda \le 2$, then the range of $\vert \lambda \vec{a}\vert $ | (IV) $(\dfrac{\vec{a} \cdot \vec{b}}{\vert \vec{b}\vert ^2}) \vec{b}$ | Choose the correct answer from the options given below:

In $\triangle ABC$ :

(A) $\vec{AB} + \vec{BC} + \vec{CA} = \vec{O}$

(B) $\vec{AB} + \vec{BC} - \vec{AC} = \vec{O}$

(C) $\vec{AB} + \vec{BC} - \vec{CA} = \vec{O}$

(D) $\vec{AB} - \vec{CB} + \vec{CA} = \vec{O}$

(E) $\vec{AB} - \vec{CB} - \vec{CA} = \vec{O}$

Choose the correct answer from the options given below :

If $\vec{a}$ is any vector, then $|\vec{a} \times \hat{i}|^2 + |\vec{a} \times \hat{j}|^2 + |\vec{a} \times \hat{k}|^2$ is equal to

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 $\vec{p}$ and $\vec{q}$ are two unit vectors such that $|\vec{p} + \vec{q}| = \sqrt{2}$, then which of the following are correct?

(A) $|\vec{p}| = |\vec{q}| = 1$

(B) $\vec{p}$ and $\vec{q}$ are orthogonal vectors

(C) $\vec{p}$ and $\vec{q}$ are collinear vectors

(D) $(4\vec{p} - \vec{q}).(2\vec{p} + \vec{q}) = 7$

Choose the correct answer from the options given below:

The area (in sq. units) of the parallelogram whose diagonals are represented by the vectors $\vec{a}=2\hat{i}-\hat{j}+\hat{k}$ and $\vec{b}=3\hat{i}+4\hat{j}-\hat{k}$, is

Let $\vec{a} = (\hat{i}+4\hat{j}+2\hat{k})$, $\vec{b} = (3\hat{i}-2\hat{j}+7\hat{k})$, $\vec{c} = (2\hat{i}-\hat{j}+4\hat{k})$. A vector $\vec{d}$, which is perpendicular to both $\vec{a}$ and $\vec{b}$, such that $\vec{c}.\vec{d} = 18$, is