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

If $\vec{a}$ and $\vec{b}$ are two non-zero vectors such that $|\vec{a}\times\vec{b}|^2 + |\vec{a}.\vec{b}|^2 = 100$ and $|\vec{a}| = 5$ then the value of $|\vec{b}|$ is

If $\vec{a} = \hat{i} - 2\hat{j} + 3\hat{k}$ and $\vec{b} = \hat{i} + 2\hat{j} + \hat{k}$, then which of the following statements are correct?

(A) $\vec{a}$ and $\vec{b}$ are perpendicular to each other.

(B) $\vec{a}$ and $\vec{b}$ are parallel.

(C) $|\vec{a}\times\vec{b}| = 2\sqrt{21}$.

(D) Projection of $\vec{a}$ on $\vec{b}$ = 0.

Choose the correct answer from the options given below:

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

Which of the following statements are true?

(A) If $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$, then $x, y, z$ are called direction ratios of $\vec{r}$.

(B) For any two vectors $\vec{a}$ and $\vec{b}$, $\vec{a} + \vec{b} = \vec{b} + \vec{a}$

(C) $\vec{a} \perp \vec{b}$ if and only if $\vec{a} \times \vec{b} = \vec{0}$

(D) Projection of $\vec{b}$ on $\vec{a}$ is $\frac{\vec{a} \cdot \vec{b}}{|\vec{a}|^2}$

Choose the correct answer from the options given below:

If $\vec{a} \cdot \vec{b} = \vec{a} \cdot \vec{c}$, $\vec{a} \times \vec{b} = \vec{a} \times \vec{c}$ and $\vec{a} \neq {0}$, then the vector $\vec{b}$ in equal to.

The unit vector perpendicular to each of the vectors $\vec{a}+\vec{b}$ and $\vec{a}-\vec{b}$, where $\vec{a}=\hat{i}+\hat{j}+\hat{k}$ and $\vec{b}=\hat{i}+2 \hat{j}+3 \hat{k}$, is :