Vectors $\vec{a}, \vec{b}, \vec{c}$ are such that $\vec{a} + \vec{b} + \vec{c} = 0$ and $|\vec{a}| = |\vec{b}| = |\vec{c}|$ then, the angle between any two of them, is

If $\vec{a}, \vec{b}$ and $\vec{c}$ are three unit vectors such that $\vec{a} + 2\vec{b} - 3\vec{c} = \vec{0}$, then the value of $2\vec{a}.\vec{b} - 6\vec{b}.\vec{c} - 3\vec{c}.\vec{a}$ is

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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Match List-I with List-II

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Let $\vec{a}, \vec{b}$ be two vectors such that $|\vec{a}| = 2, |\vec{b}| = 3, \vec{a} \cdot \vec{b} = 4$, then $|\vec{a} - \vec{b}|$ is equal to

If $\vec{a}$ and $\vec{b}$ are two unit vectors inclined at an angle $\theta$, then the value of $|\vec{a}-\vec{b}|$ is

If $\vec{a}$ and $\vec{b}$ are two non-zero vectors such that $|\vec{a} \cdot \vec{b}| = |\vec{a} \times \vec{b}|$, then the angle $\theta$ between $\vec{a}$ and $\vec{b}$ is