Given $A = 2 \hat{i} + \hat{j}$,$B = 3 \hat{j} - \hat{k}$,and $C = 6 \hat{i} - 2 \hat{k}$. The value of $A - 2B + 3C$ is:

The $x$ and $y$ components of vector $\vec{P}$ have magnitudes $1$ and $3$,and the $x$ and $y$ components of the resultant of $\vec{P}$ and $\vec{Q}$ have magnitudes $5$ and $6$ respectively. What is the magnitude of $\vec{Q}$?

$A$ truck travelling due north at $20 \, m/s$ turns west and travels at the same speed. The change in its velocity is

The magnitudes of two vectors are $8$ units and $6$ units respectively. Find the magnitude of the resultant vector if the angle between these two vectors is $(i) \theta = 0^{\circ}$,$(ii) \theta = 180^{\circ}$,$(iii) \theta = 90^{\circ}$,and $(iv) \theta = 120^{\circ}$.

The vectors $(\vec{A} + \vec{B})$ and $(\vec{A} - \vec{B})$ are perpendicular to each other. This is possible under the condition:

$\overrightarrow{A}$ and $\overrightarrow{B}$ are two vectors of equal magnitudes and $\theta$ is the angle between them. The angle between $\overrightarrow{A}$ or $\overrightarrow{B}$ with their resultant is