Vector Operations>Multiplication of a Vector by a Scalar

Decompose the vector \[6 \hat{i} - 3 \hat{j} - 6 \hat{k}\] into vectors which are parallel and perpendicular to the vector \[\hat{i} + \hat{j} + \hat{k} .\] 

Find the values of x and y if the vectors \[\vec{a} = 3 \hat{i} + x \hat{j} - \hat{k} \text{ and } \vec{b} = 2 \hat{i} + \hat{j} + y \hat{k}\] are mutually perpendicular vectors of equal magnitude. 

Find \[\left| \vec{a} \right| and \left| \vec{b} \right|\] if 

\[\left( \vec{a} + \vec{b} \right) \cdot \left( \vec{a} - \vec{b} \right) = 3\text{  and } \left| \vec{a} \right| = 2\left| \vec{b} \right|\]

If `|vec"a"| = 4, |vec"b"| = 3` and `vec"a".vec"b" = 6 sqrt(3)`, then find the value of `|vec"a" xx vec"b"|`.

The value of is `hati.(hatj xx hatk)+hatj.(hatixxhatk)+hatk.(hatixxhatj)` is ______.

If a vector \[\vec{a}\] is perpendicular to two non-collinear vectors \[\vec{b} \text{ and } \vec{c} , \text{ then show that } \vec{a}\] is perpendicular to every vector in the plane of \[\vec{b} \text{ and } \vec{c} .\] 

If \[\vec{a} \cdot \vec{a} = 0 \text{ and } \vec{a} \cdot \vec{b} = 0,\] what can you conclude about the vector \[\vec{b}\] ?

Let \[\vec{a} = 5 \hat{i} - \hat{j} + 7 \hat{k} \text{ and } \vec{b} = \hat{i} - \hat{j} + \lambda \hat{k} .\] Find λ such that \[\vec{a} + \vec{b}\] is orthogonal to \[\vec{a} - \vec{b}\]