State whether the following statements are true or false:
$(a)$ The unit vectors ${\hat i}$ and ${\hat j}$ along the $x$ and $y$-axes change with time.
$(b)$ If the angle between $\overrightarrow A$ and $\overrightarrow B$ is ${\theta _1}$ and the angle between $\overrightarrow A$ and $\overrightarrow C$ is ${\theta _2}$,then $\overrightarrow A \cdot \overrightarrow B = \overrightarrow A \cdot \overrightarrow C$ implies $\overrightarrow B = \overrightarrow C$.
$(c)$ The resultant vector of two coplanar vectors is also a coplanar vector.

$A$ spherical raindrop evaporates at a rate proportional to its surface area. The differential equation involving the rate of change of its radius $r$ with time $t$ is ... (where $k$ is a positive constant)

$\overrightarrow{A} = 4\hat{i} + 3\hat{j}$ and $\overrightarrow{B} = 4\hat{i} + 2\hat{j}$. Find a vector parallel to $\overrightarrow{A}$ but with a magnitude five times that of $\overrightarrow{B}$.

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

Read each statement below carefully and state with reasons,if it is true or false:
$(a)$ The magnitude of a vector is always a scalar.
$(b)$ Each component of a vector is always a scalar.
$(c)$ The total path length is always equal to the magnitude of the displacement vector of a particle.
$(d)$ The average speed of a particle (defined as total path length divided by the time taken to cover the path) is either greater or equal to the magnitude of average velocity of the particle over the same interval of time.
$(e)$ Three vectors not lying in a plane can never add up to give a null vector.

How is the magnitude of a vector quantity represented?