Multiplication of a Vector by a Scalar

If \[\vec{a}\] is a unit vector, then find \[\left| \vec{x} \right|\]  in each of the following. 

\[\left( \vec{x} - \vec{a} \right) \cdot \left( \vec{x} + \vec{a} \right) = 12\] 

If \[\left| \vec{a} \right| = a \text{ and } \left| \vec{b} \right| = b,\] prove that \[\left( \frac{\vec{a}}{a^2} - \frac{\vec{b}}{b^2} \right)^2 = \left( \frac{\vec{a} - \vec{b}}{ab} \right)^2 .\] 

Compute the magnitude of the following vector:

`veca = hati + hatj + hatk;` `vecb = 2hati - 7hatj - 3hatk`;  `vecc = 1/sqrt3 hati + 1/sqrt3 hatj - 1/sqrt3 hatk`

If \[\vec{c}\] s perpendicular to both \[\vec{a} \text{ and } \vec{b}\] then prove that it is perpendicular to both \[\vec{a} + \vec{b} \text{ and } \vec{a} - \vec{b}\] 

If θ is the angle between any two vectors `veca` and `vecb,` then `|veca.vecb| = |veca xx vecb|` when θ is equal to ______.

If \[\vec{a}\] is a unit vector, then find \[\left| \vec{x} \right|\]  in each of the following. 

\[\left( \vec{x} - \vec{a} \right) \cdot \left( \vec{x} + \vec{a} \right) = 8\] 

If \[\vec{a} + \vec{b} + \vec{c} = \vec{0} ,\] show that the angle θ between the vectors \[\vec{b} \text{ and } \vec{c}\] is given by  \[\frac{\left| \vec{a} \right|^2 - \left| \vec{b} \right|^2 - \left| \vec{c} \right|^2}{2\left| \vec{b} \right| \left| \vec{c} \right|} .\]