Find all vectors of magnitude 10\(\sqrt { 3 }\)   that are perpendicular to the plane of:
\(\overset { \wedge }{ i } +2\overset { \wedge }{ j } +\overset { \wedge }{ k } \ and\ -\overset { \wedge }{ i } +3\overset { \wedge }{ j } +4\overset { \wedge }{ k } \) 

Using vector, find the value of 'k' such that the point: (k, -10, 3), (1, -1, 3) and (3, 5, 3) are collinear.

If A,B,C are position vectors:  \(\hat{i}+\hat{j}-\hat{k}, 2 \hat{i}-\hat{j}+3 \hat{k}, \hat{i}-2 \hat{j}+\hat{k}\)
Respectively, find the projection of \(\overset { \rightarrow }{ AB } \) along \(\overset { \rightarrow }{ CD } \).

If \(\overset { \rightarrow }{ a } \) and \(\overset { \rightarrow }{ b } \) are perpendicular vectors, \(|\overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } |=13\) and \(|\overset { \rightarrow }{ a }|\) = 5, then find the value of  |\(\overset { \rightarrow }{ b } \)|.

If   \(\overset { \rightarrow }{ a } \) and \(\overset { \rightarrow }{ b } \) are two unit vectorssuch that \(\overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \)  is also a unit vector, then find the angle between  \(\overset { \rightarrow }{ a } \) and \(\overset { \rightarrow }{ b } \).

if \(\overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } =0\) and \(\overset { \rightarrow }{ |a| } =3,\overset { \rightarrow }{ |b| } =7\ and\ \overset { \rightarrow }{ |c| } =7\), Find the angle between \(\overset { \rightarrow }{ a } \) and \(\overset { \rightarrow }{ b } \).

Find a vector \(\overset { \rightarrow }{ a } \) of magnitude \(5\sqrt { 2 } \)  making an angle \(\pi\over4\) with x-axis ,\(\pi\over2\) with y-axis and an angle '\(\theta\)' with z-axis

Lagrange's identify prove that : \({( }{ \overrightarrow { a } *\overrightarrow { b) } }^{ 2 }=\overset { \rightarrow }{ { |a| }^{ 2 } } \overset { \rightarrow }{ { |b| }^{ 2 } } -{ ( }{ \overrightarrow { a } .\overrightarrow { b) } }^{ 2 }\)

Find the volume of parallelopiped whose sides are given by vectors: \(2\overset { \wedge }{ i } -3\overset { \wedge }{ j } +4\overset { \wedge }{ k } ,\overset { \wedge }{ i } +2\overset { \wedge }{ j } -\overset { \wedge }{ k } and3\overset { \wedge }{ i } -\overset { \wedge }{ j } +2\overset { \wedge }{ k } \)

Show that the four points A(4, 5, 1), B(0, -1, -1), C(3, 9, 4) and D(-4, 4, 4) are coplanar.