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

If $\vec{a}$ and $\vec{b}$ are two vectors such that $|\vec{a}| = 10, |\vec{b}| = 2$ and $\vec{a} \cdot \vec{b} = 12$, then $|\vec{a} \times \vec{b}|$ is equal to

Which of the following statements are correct ?

A. If $\vec{a}$ and $\vec{b}$ are two adjacent sides of a triangle, then the area of the triangle is $\frac{1}{2}|\vec{a}\times\vec{b}|$

B. If $\vec{a}$ and $\vec{b}$ are two adjacent sides of a triangle, then the area of the triangle is $\frac{1}{2}|\vec{a}.\vec{b}|$

C. If $\vec{a}$ and $\vec{b}$ are two adjacent sides of a parallelogram, then its area is $|\vec{a}\times\vec{b}|$

D. If $\vec{a}$ and $\vec{b}$ are two adjacent sides of a parallelogram, then its area is $|\vec{a}.\vec{b}|$

Choose the correct answer from the options given below:

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 :

If $|\vec a|=2, |\vec b|=7, \vec a \times \vec b = 3\hat i + 2\hat j + 6\hat k$, then the angle between $\vec a$ and $\vec b$ is equal to

If $P(\vec{r})$ is a position vector of a point with scalar components $(x,y,z)$, then the value of $(\vec{r}.\hat{i})(-\hat{i})-(\vec{r}.\hat{j})\hat{j}+(\vec{r}.(-\hat{k}))\hat{k}$, is:

Let $\vec{a},\vec{b},\vec{c}$ are three vectors such that $|\vec{a}|=3, |\vec{b}|=2, |\vec{c}|=1$. If the projection of $\vec{b}$ along $\vec{c}$ is equal to the projection of $\vec{a}$ along $\vec{c}$ and $\vec{b} \perp \vec{a}$ , then the value of $|2\vec{a}-2\vec{b}+3\vec{c}|$ is equal to