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 $\vec{a}$ 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 $\vec{a},\vec{b}$ and $\vec{c}$ are three vectors such that $\lvert \vec{a} \rvert=\frac{1}{\sqrt{2}}$, $\lvert \vec{b} \rvert=\frac{1}{\sqrt{3}}$, $\lvert \vec{c} \rvert=\frac{1}{\sqrt{6}}$, $\lvert \vec{a}+\vec{b}+\vec{c} \rvert=1$ and $\vec{c}=\lambda(\vec{a}\times\vec{b})$. The angle between $\vec{a}$ and $\vec{b}$ 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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In $\triangle ABC$, Which of the following are True?

A. $\overline{AB}+\overline{BC}+\overline{CA}=\vec{0}$

B. $\overline{AB}+\overline{BC}-\overline{AC}=\vec{0}$

C. $\overline{AB}+\overline{BC}-\overline{CA}=\vec{0}$

D. $\overline{AB}-\overline{CB}+\overline{CA}=\vec{0}$

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Match List - I with List - II.

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Two vectors $\vec{a}$ and $\vec{b}$ are inclined at angle $\theta=60°$ such that $|\vec{a}|=1$, $|\vec{b}|=2$, then $\left|(\vec{a}+3\vec{b})\times(3\vec{a}-\vec{b})\right|^2$ is equal to