Two vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ are defined as $\overrightarrow{A}=a\hat{\mathrm{i}}$ and $\overrightarrow{B}=a\left(\mathrm{cos\omega t}\hat{\mathrm{i}}+\mathrm{sin\omega t}\hat{\mathrm{j}}\right)$ where $a$ is a constant and $\omega =\frac{\pi }{6} \mathrm{rad} {\mathrm{s}}^{-1}.$ If $|\overrightarrow{A}+\overrightarrow{B}|=\sqrt{3}|\overrightarrow{A}-\overrightarrow{B}|$ at time $t=\tau$ for the first time, the value of $\tau ,$ in seconds is ______.

The first drawing shows three displacement vectors $\overrightarrow{A}$, $\overrightarrow{B}$ and $\overrightarrow{C}$ which are added in a tail-to-head fashion. The resultant vector is labelled $\overrightarrow{R}$. Which of the following drawings shows the correct resultant vector for $\overrightarrow{A}+\overrightarrow{B}-\overrightarrow{C}$?

The given figure shows the sum of three displacement vectors, $\overrightarrow{A}$, $\overrightarrow{B}$ and $\overrightarrow{C}$. The resultant vector is $\overrightarrow{R}$. Which of the following figures show the correct resultant vector for $\overrightarrow{A}-\overrightarrow{B}-\overrightarrow{C}$?

Which of the following displacement vectors (if any) are equal?

Consider the following four force vectors:

(a) ${\overrightarrow{F}}_1=50.0 \mathrm{N}$, due east

(b) ${\overrightarrow{F}}_2=10.0 \mathrm{N}$, due east

(c) ${\overrightarrow{F}}_3=40.0 \mathrm{N}$, due west

(d) ${\overrightarrow{F}}_4=30.0 \mathrm{N}$, due west

Which two vectors add together to give a resultant with the smallest magnitude and which two vectors add to give a resultant with the largest magnitude? In each case, specify the magnitude and direction of the resultant.

A car is being pulled out of the mud by two forces that arc applied by the two ropes shown in the drawing. The dashed line in the drawing bisects the $90°$ angle. The magnitude of the force applied by each rope is $2000 \mathrm{N}$. Arrange the force vectors tail to head and use the graphical technique to answer the following questions:

How would the single rope be directed relative to the dashed line?

A car is being pulled out of the mud by two forces that are applied by the two ropes shown in the drawing. The dashed line in the drawing bisects the ${90}^0$ angle. The magnitude of the force applied by each rope is $2000 \mathrm{N}$. Arrange the force vectors tail to head and use the graphical technique to answer the following questions:

(a) How much force would a single rope need to apply to accomplish the same effect as the two forces are added together?

(b) How would the single rope be directed relative to the dashed line?