Which of the following statements is/are TRUE ?

A. The sine of the angle between the vectors $\vec{a}=\hat{i}+3\hat{j}-2\hat{k}$ and $\vec{b}=-2\hat{i}+2\hat{j}-4\hat{k}$ is $\frac{2}{\sqrt{7}}$

B. The area of the parallelogram PQRS is $\frac{1}{2}|\vec{PR}\times\vec{QS}|$

C. If $\vec{a}=5\hat{i}-4\hat{j}+3\hat{k}$ then $|\vec{a}\times\hat{i}|^2+|\vec{a}\times\hat{j}|^2+|\vec{a}\times\hat{k}|^2=100$

D. Vectors perpendicular to the vectors $\vec{a}=2\hat{i}-\hat{j}+2\hat{k}$ and $\vec{b}=-\hat{i}+4\hat{j}-3\hat{k}$ are given by $\lambda(-\hat{i}+2\hat{j}+2\hat{k})$ where $\lambda$ is a scalar.

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The projection of the vector $\vec{a} = \hat{i} + 2\hat{j} - 3\hat{k}$ on the vector $2\hat{i} + 6\hat{j} + 3\hat{k}$ is

If $\vec{a} = \hat{i} + \hat{k}$, $\vec{b} = \hat{j} - \hat{k}$ and $\vec{c} = \hat{i} + \hat{j} + \hat{k}$ such that $\vec{r} \times \vec{b} = \vec{c} \times \vec{b}$ and $\vec{r} \cdot \vec{a} = 0$, then $\vec{r}$ is:

Let $\vec{a}$ and $\vec{b}$ be two vectors, such that $|\vec{a}| = 3, |\vec{b}| = 5$ and the angle between them is $60°$, then the value of $|\vec{a} + \vec{b}|$ is

If $\hat{i}$, $\hat{j}$ and $\hat{k}$ are unit vectors along the co-ordinate axes OX, OY and OZ respectively, then

(A) $\hat{i} \times \hat{j} = \hat{k}$

(B) $\hat{k} \times \hat{i} = -\hat{j}$

(C) $\hat{j} \cdot \hat{j} = 1$

(D) $\hat{j} \cdot \hat{k} = 0$

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The angle between two lines whose direction ratios are propotional to $1,1,-2$ and $(\sqrt{3}-1),(-\sqrt{3}-1),-4$ is :

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

(A) unit vector along $\vec{a}$ is $\frac{1}{3}(2\hat{i}+2\hat{j}-\hat{k})$

(B) $|3\vec{a}| = 9$

(C) $\vec{a}$ is perpendicular to $2\hat{i}+3\hat{j}+2\hat{k}$

(D) Direction Ratios of $\vec{a}$ are 2,-2,1

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