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:

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

Let us consider vector $\vec{a} = (c\log_2 x)\hat{i} - 6\hat{j} + 3\hat{k}$ and $\vec{b} = (\log_2 x)\hat{i} + 2\hat{j} + (2c\log_2 x)\hat{k}$, then which of following statements are TRUE?

(Where $x \in \mathbb{R}$ (set of real numbers))

(A) Vector $\vec{a}$ and $\vec{b}$ make obtuse angle for any $x \in (0,\infty)$ if $c \in (-1,0)$

(B) Vector $\vec{a}$ and $\vec{b}$ makes obtuse angle for any $x \in \mathbb{R}$ if $c > 0$

(C) Vector $\vec{a}$ and $\vec{b}$ makes obtuse angle for any $x \in (0,\infty)$ if $c \in \left(-\frac{4}{3}, -1\right]$

(D) Vectors $\vec{a}$ and $\vec{b}$ makes obtuse angle for $x \in \mathbb{R}$ if $c \in (1,2)$

Choose the correct answer from the options given below:

The area of triangle with vertices P, Q, R is given by (where $\vec{AB}$ = position vector of point B – position vector of point A)

If $\vec a=\hat i+\hat j+2\hat k$ and $\vec b=2\hat i+\hat j+2\hat k$, then the unit vector in the direction of $2\vec a-\vec b$ is equal to :

Let $\vec{a} = 2\hat{i} - \hat{j}, \vec{b} =- 4\hat{j} + k\,\text{and}\,\vec{c} = \hat{i} + 2\hat{k}$. If $\vec{d}$ is a vector perpendicular to both $\vec{a}$ and $\vec{b}$ such that $\vec{c} \cdot \vec{d} = 34$, then $|\vec{d}|$ is equal to

If $\vec{a}, \vec{b}$ and $\vec{c}$ are three vectors such that $\vec{a} + \vec{b} + \vec{c} = \vec{0}$, $|\vec{a}| = 7$, $|\vec{b}| = 3$ and $|\vec{c}| = 5$, then angle between $\vec{b}$ and $\vec{c}$ is