If $|\vec{A} + \vec{B}| = |\vec{A}| + |\vec{B}|$,the angle between $\vec{A}$ and $\vec{B}$ is ....... $^o$.

State with reasons,whether the following algebraic operations with scalar and vector physical quantities are meaningful:
$(a)$ adding any two scalars,
$(b)$ adding a scalar to a vector of the same dimensions,
$(c)$ multiplying any vector by any scalar,
$(d)$ multiplying any two scalars,
$(e)$ adding any two vectors,
$(f)$ adding a component of a vector to the same vector.

If water at $100^\circ C$ cools in $10$ minutes to $80^\circ C$ and to $65^\circ C$ in the next $10$ minutes, then the room temperature will be ...

The resultant of two vectors $A$ and $B$ makes an angle $\alpha$ with $A$ and $\beta$ with $B$. Which of the following is correct?

The position vectors of points $A, B, C$ and $D$ are $A = 3\hat{i} + 4\hat{j} + 5\hat{k}$,$B = 4\hat{i} + 5\hat{j} + 6\hat{k}$,$C = 7\hat{i} + 9\hat{j} + 3\hat{k}$ and $D = 4\hat{i} + 6\hat{j}$. Then the displacement vectors $\overrightarrow{AB}$ and $\overrightarrow{CD}$ are:

The values of $x$ and $y$ for which vectors $\vec{A} = (6 \hat{i} + x \hat{j} - 2 \hat{k})$ and $\vec{B} = (5 \hat{i} + 6 \hat{j} - y \hat{k})$ are parallel are