Addition and Subtraction of Vectors

A bus is moving on a straight road towards north with a uniform speed of  $50 \mathrm{km} {\mathrm{h}}^{-1}$  turns through  $90°$ anticlockwise. If the speed remains unchanged after turning , the increase in the velocity of bus in the turning process is:

If two forces of magnitude $4 \mathrm{kN}$ and $8 \mathrm{kN}$ act on a body, then their minimum resultant is 

If  $\left(\stackrel{\rightharpoonup }{V_1}\right) = 3\widehat{i } +4\widehat{j }+\hat{k} and \left(\stackrel{\rightharpoonup }{V_2}\right) = \widehat{i } -\widehat{j }-\hat{k}$then determine magnitude of $\left(\stackrel{\rightharpoonup }{V_1}\right) +\left(\stackrel{\rightharpoonup }{V_2}\right)$.

The vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ are two mutually perpendicular vectors. Their dot product remains the same when the magnitude of $\overrightarrow{A}$ becomes three times its previous value.

How many minimum numbers of vectors of unequal magnitudes are required to give zero resultant?

The square of resultant of two equal forces is three times their product. Angle between the forces is

Let the vectors $\overrightarrow{PQ}, \overrightarrow{QR}, \overrightarrow{RS}, \overrightarrow{ST}, \overrightarrow{TU}$ and $\overrightarrow{UP}$ represent the sides of a regular hexagon.

Statement I: $\overrightarrow{PQ}\times \left(\overrightarrow{RS}+\overrightarrow{ST}\right)\ne \overrightarrow{0}$

Statement II: $\overrightarrow{PQ}\times \overrightarrow{RS}=\overrightarrow{0}$ and $\overrightarrow{PQ}\times \overrightarrow{ST}\ne \overrightarrow{0}$

If $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ are unit vectors, then ${\left|\overrightarrow{a}-\overrightarrow{b}\right|}^2+{\left|\overrightarrow{b}-\overrightarrow{c}\right|}^2+{\left|\overrightarrow{c}-\overrightarrow{a}\right|}^2$ does not exceed.