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 A and B are two points vectors and respectively. write the position vectors of a point of a P, which divides the line segment AB internally in the ratio 1 : 2

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 \(\overset { \rightarrow }{ b } \).

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: \(2\overset { \wedge }{ i } -3\overset { \wedge }{ j } +4\overset { \wedge }{ k } ,\overset { \wedge }{ i } +2\overset { \wedge }{ j } -\overset { \wedge }{ k } and3\overset { \wedge }{ i } -\overset { \wedge }{ j } +2\overset { \wedge }{ k } \)

Prove that:\(\left[ \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } ,\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } ,\overset { \rightarrow }{ c } +\overset { \rightarrow }{ a } ]=2[\overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] \)

Show that the vector \(\overset { \rightarrow }{ a } \), \(\overset { \rightarrow }{ b } \) and \(\overset { \rightarrow }{ c } \) are coplanar if \(\overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } ,\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } \ and \ \overset { \rightarrow }{ c } +\overset { \rightarrow }{ a } \) are coplanar.