$A$ force,$\vec{F}=(4 \hat{i}+3 \hat{j}-5 \hat{k}) \text{ N}$,is acting on a body. If the horizontal direction is taken along the $\hat{i} + \hat{j}$ direction,find the angle $\theta$ that the force makes with this horizontal direction.

Given: $\vec{A} = 2\hat{i} + p\hat{j} + q\hat{k}$ and $\vec{B} = 5\hat{i} + 7\hat{j} + 3\hat{k}$. If $\vec{A} \parallel \vec{B}$,then the values of $p$ and $q$ are,respectively:

In a regular octagon $ABCDEFGH$ with center $O$,what is the sum of $\overrightarrow{ AB }+\overrightarrow{ AC }+\overrightarrow{ AD }+\overrightarrow{ AE }+\overrightarrow{ AF }+\overrightarrow{ AG }+\overrightarrow{ AH }$ if $\overrightarrow{ AO }=2 \hat{ i }+3 \hat{ j }-4 \hat{ k }$?

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.

Find the angle between $\vec{A} = 3\hat{i} - \hat{j} + 4\hat{k}$ and the $Z$-axis.

What should be the angle $\theta$ between two vectors $\vec{A}$ and $\vec{B}$ so that the magnitude of the resultant vector $\vec{R}$ is maximum (in $^{\circ}$)?