2021 HCI P1 Q7

It is given that \(\mathbf{a}\) and \(\mathbf{b}\) are non zero vectors.

1. Given that \(\left| \mathbf{a}+\mathbf{b} \right|=\left| \mathbf{a}-\mathbf{b} \right|\), by considering suitable scalar product, comment on the relationship between \(\mathbf{a}\) and \(\mathbf{b}\).[2]
2. Given that \(\frac{\mathbf{a}}{\left| \mathbf{a} \right|}=\frac{\mathbf{b}}{\left| \mathbf{b} \right|}\), what can you say about the relationship between \(\mathbf{a}\) and \(\mathbf{b}\)?[1]
3. Given that \({2\mathbf{a} + \mathbf{b} = -5\mathbf{j} + \mathbf{k}}\), \({\mathbf{c} = \mathbf{i} - 2\mathbf{j} + 2\mathbf{k}}\), \({\mathbf{a}.\mathbf{c} = 5}\) and \({\mathbf{b}}\) is a unit vector,
   1. find the value of \(\mathbf{b}.\mathbf{c}\).[2]
   2. find the sine of the angle between \(\mathbf{b}\) and \(\mathbf{c}\).[2]
   3. find \(\left| \mathbf{b}\times \mathbf{c} \right|\), and state the geometrical meaning of this result.[2]