If \(\overrightarrow{a},\overrightarrow{b}\) are unit vectors and \(\theta\) is the angle between them, show that \(cos {\theta \over 2}={1\over2}|\overrightarrow{a}+\overrightarrow{b}|\)

If \(\overrightarrow{a}=-3\hat{i}+4\hat{j}-7\hat{k}\) and \(\overrightarrow{b}=6\hat{i}+2\hat{j}-3\hat{k},\) verify \(\overrightarrow{a}\) are \(\overrightarrow{a}\times \overrightarrow{b}\) perpendicular to each other.

Find the unit vectors perpendicular to each of the vectors \(\overrightarrow{a}+\overrightarrow{b}\) and \(\overrightarrow{a}-\overrightarrow{b}\), where \(\overrightarrow{a}=\hat{i}+\hat{j} +\hat{k} \) and \(\overrightarrow{b} =\hat{i}+2\hat{j} +3\hat{k} \).

If \({1\over2},{1\over \sqrt{2}}\), a are the direction cosines of some vector, then find a.

Find the unit vector in the direction of the vector \(\overrightarrow { a } -2\overrightarrow { b } +3\overrightarrow { c } \) if \(\overrightarrow { a } =\hat { i } +\hat { j } ,\overrightarrow { b } =\hat { j } +\hat { k } \) and \(\overrightarrow { c } =\hat { i } +\hat { k } \) .

Find the angle A of the triangle whose vertices are A(0, -1, 2), B(3, 1, 4) and C(5, 7, 1).

Find the real value of \(\lambda \) so that the vectors \(\overrightarrow { a } =\hat { i } +\hat { j } +\lambda \hat { k } \) and \(\overrightarrow { b } =2\hat { i } +\lambda \hat { k } \)  are perpendicular.

Show that each of the given three vectors is a unit vector. \(\frac { 1 }{ 7 } (2\hat { i } +3\hat { j } +6\hat { k } );\frac { 1 }{ 7 } (3\hat { i } -6\hat { j } +2\hat { k } );\frac { 1 }{ 7 } (6\hat { i } +2\hat { j } -3\hat { k } )\) Also, show that they are mutually perpendicular to each other.

Find \(|\overrightarrow { x } |\) if for a unit vector \(\overrightarrow { a } ,(\overrightarrow { x } -\overrightarrow { a } ).(\overrightarrow { x } +\overrightarrow { a } )=12\)

Let \(\overrightarrow { a } ,\overrightarrow { b } \) and \(\overrightarrow { c } \)  be non-coplanar vectors. Let A, B and C be the points whose position vectors with respect to the origin O are \(\overrightarrow { a } +2\overrightarrow { b } +3\overrightarrow { c } ,-2\overrightarrow { a } +3\overrightarrow { b } +5\overrightarrow { c } \) and \(7\overrightarrow { a } -\overrightarrow { c } \)  respectively. Then prove that A, B and C are collinear.