Classify the following on scalar and vector quantities:
(i) Work
(ii) Force
(iii) Velocity
(iv) Displacement.

In a triangle ABC, Show that \(\overset\rightarrow {AB}+\overset\rightarrow {BC}+\overset\rightarrow {CA}=0\)

Find the unit vector in the direction of \(\overset\rightarrow a+\overset\rightarrow b\)if \(\overset\rightarrow a= 2\overset\wedge i+\overset\wedge j+3\overset\wedge k\), and \(\overset\rightarrow b= \overset\wedge i+2\overset\wedge j-\overset\wedge k\)

Find the position vector of c which divides the line segment joining A & B whose position vectors are \(3\overset\rightarrow a+\overset\rightarrow b\)and \(\overset\rightarrow a-3\overset\rightarrow b\) internally in the ratio 2:3.

Find the direction cosines of the vector joining the points A(1, 2, - 3) and B(- 1, - 2, 1) directed from B to A.

If \(\overset\rightarrow a=3\overset\wedge i+7\overset\wedge j+2\overset\wedge k,\overset\rightarrow b=7\overset\wedge i-2\overset\wedge j+3\overset\wedge k\). Is \(\left|\overset\rightarrow a \right| =\left|\overset\rightarrow b \right| \).Can we say \(\overset\rightarrow a=\overset\rightarrow b?\)Give reason.

If \(\left|\overset\rightarrow a+\overset\rightarrow b \right| =60,\left|\overset\rightarrow a-\overset\rightarrow b \right|=40\)and \(\left| \overset\rightarrow a \right| =22,\) then find \(\left| \overset\rightarrow b \right| \).

If \(\overset\rightarrow a\) and \(\overset\rightarrow b\) are two perpendicular vector then show that \((\overset\rightarrow a+\overset\rightarrow b)^{2}=(\overset\rightarrow a-\overset\rightarrow b)^{2}\)

Find the projection of \(\overset\rightarrow a+\overset\rightarrow b\) on \(\overset\rightarrow a-\overset\rightarrow b,\)\(\overset\rightarrow a=i+2j+k,\overset\rightarrow b=3\overset\wedge i+\overset\wedge j-\overset\wedge k\).

If two vectors \(\overset\rightarrow a\) and \(\overset\rightarrow b\) are such that \(\left| \overset { \rightarrow }{ a } \right| =3,\left| \overset { \rightarrow }{ b } \right| =1\)and \(\overset\rightarrow a.\overset\rightarrow b=2\). Find \((2\overset\rightarrow a-3\overset\rightarrow b).(3 \overset\rightarrow a+\overset\rightarrow b).\)

Find \(\left| \overrightarrow { a} \times \overrightarrow { b } \right| ,\)if \(\overrightarrow a=\overset\wedge i+2\overset\wedge j-\overset\wedge k,\overrightarrow b=3\overset\wedge i+\overset\wedge j-\overset\wedge k\)

Find the area of parallelogram whose adjacent sides are determined by the vector.
\(\overset\rightarrow a=\overset\wedge i-\overset\wedge j+2\overset\wedge k\) and \(\overset\rightarrow b=2\overset\wedge i-\overset\wedge j-\overset\wedge k\)

Find \(\left| \overrightarrow {a } \times \overrightarrow { b } \right| \) if \(\left| \overrightarrow { a} \right| =10,\left| \overrightarrow { b } \right| =2\) and \(\overrightarrow a.\overrightarrow b=12\).

Find the magnitude of two vectors \(\overrightarrow a\) and \(\overrightarrow b\) of equal magnitude such that the angle between them is a 60^o and their scalar product is \(\frac{1}{2}.\)

Prove that \(\left| \overrightarrow { a } \times \overrightarrow { b } \right| ^{ 2 }=\begin{vmatrix} \overrightarrow { a } .\overrightarrow { a } & \overrightarrow { a } .\overrightarrow { b } \\ \overrightarrow { a } .\overrightarrow { b } & \overrightarrow { b } .\overrightarrow { b } \end{vmatrix}\)