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

Give three properties which have different magnitudes and also have different directions.

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.

The resultant magnitude of two vectors of same magnitude is equal to the magnitude of either. The angle between the two vectors is (in $^{\circ}$)

The expression $\left( \frac{1}{\sqrt{2}}\hat{i} + \frac{1}{\sqrt{2}}\hat{j} \right)$ is a

Find $|\Delta \overrightarrow{a}|$ and $\Delta a$ when a vector $\overrightarrow{a}$ is rotated by an angle $d\theta$.