Find λ so that the vectors λ so that the vectors \(2\hat { i } +\lambda \hat { j } +\hat { k } \) and \(\hat { i } -2\hat { j } +\hat { k } \) are perpendicular to each other.

If \(\vec { a } ,\vec { b } ,\vec { c } \) are three mutually perpendicular unit vectors, then prove that \(|\vec { a } +\vec { b } +\vec { c } |=\sqrt { 3 } \) 

If \(\vec { a } ,\vec { b } \) are any two vectors, then prove that \(\left| \vec { a } \times \vec { b } \right| ^{ 2 }+\left( \vec { a } .\vec { b } \right) ^{ 2 }=\left| \vec { a } \right| ^{ 2 }\left| \vec { b } \right| ^{ 2 }\) 

Find the vectors of magnitude 6 which are perpendicular to both the vectors \(4\vec { i } -\vec { j } +3\vec { k } \) and \(-2\vec { i } +\vec { j } -2\vec { k } \)

Find the angle between two vectors \(\vec { a } \) and \(\vec { b } \) if \(\left| \vec { a } \times \vec { b } \right| =\vec { a } .\vec { b } \)