The area of the parallelogram having diagonals \(\overset { \rightarrow }{ a } =\overset { \wedge }{ 3i } +\overset { \wedge }{ j } -2\overset { \wedge }{ k } \) and \(\overset { \rightarrow }{ b } =\overset { \wedge }{ i } -3\overset { \wedge }{ j } +4\overset { \wedge }{ k } \) is _______________

If the vector \(\overset { \wedge }{ i } +\overset { \wedge }{ j } +\overset { \wedge }{ 2k } \), \(\overset { \wedge }{ -i } +\overset { \wedge }{ 2k } \) and \(2\overset { \wedge }{ i } +x\overset { \wedge }{ j } -y\overset { \wedge }{ k } \)  are mutually orthogonal, then the values of x, y, z are _________

If \(\overset { \rightarrow }{ a } \) and \(\overset { \rightarrow }{ b } \) are two unit vectors, then the vectors \(\left( \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \right) \times \left( \overset { \rightarrow }{ a } \times \overset { \rightarrow }{ b } \right) \) is parallel to the vector ___________

The vector, d\(\overset { \wedge }{ i } +\overset { \wedge }{ j } +2\overset { \wedge }{ k } ,\overset { \wedge }{ i } +\lambda \overset { \wedge }{ j } -\overset { \wedge }{ k } \) t and \(2\overset { \wedge }{ i } -\overset { \wedge }{ j } +\lambda \overset { \wedge }{ k } \) are co-planar if _____________

The vector equation \(\vec { r } =(\hat { i } -2\hat { j } -\hat { k } )+t(6\hat { j } -\hat { k) } \) represents a straight line passing through the points

The angle between the line \(\vec { r } =(\hat { i } +2\hat { j } -3\hat { k } )+t(2\hat { i } +\hat { j } -2\hat { k } )\) and the plane \(\vec { r } .(\hat { i } +\hat { j } )+4=0\) is

If \(\vec { a } \times (\vec { b } \times \vec { c } )=(\vec { a } \times \vec { b } )\times \vec { c } \) where \(\vec { a } ,\vec { b } ,\vec { c } \) are any three vectors such that \(\vec{b} \cdot \vec{c} \neq 0 \text { and } \vec{a} \cdot \vec{b} \neq 0\), then \(\vec { a } \) and \(\vec { c } \) are

If the volume of the parallelepiped with \(\vec { a } \times \vec { b } ,\vec { b } \times \vec { c } ,\vec { c } \times \vec { a } \)  as coterminous edges is 8 cubic units, then the volume of the parallelepiped with \((\vec { a } \times \vec { b } )\times (\vec { b } \times \vec { c } ),(\vec { b } \times \vec { c } )\times (\vec { c } \times \vec { a } )\) and \((\vec { c } \times \vec { a } )\times (\vec { a } \times \vec { b } )\)as coterminous edges is,

If \(\vec { a } \) and \(\vec { b } \) are unit vectors such that \([\vec { a } ,\vec { b },\vec { a } \times \vec { b } ]=\frac { 1}{ 4 } \), then the angle between \(\vec { a } \) and \(\vec { b } \) is

If \(\vec{a}\) and \(\vec{b}\) are parallel vectors, then \([\vec { a } ,\vec { c } ,\vec { b } ]\) is equal to