1. For the vectors \(\bm{u} = \langle 1,3 \rangle\) and \(\bm{v} = \langle -4,2 \rangle,\) calculate \(\bm{u}\cdot\bm{v}.\)
2. For the vectors \(\bm{u} = \langle 5,4 \rangle\) and \(\bm{v} = \langle 1,-3 \rangle,\) calculate \(\bm{u}\cdot\bm{v}.\)
3. If \(\bm{v} = \langle 3,2 \rangle\) and \(\bm{u}\cdot\bm{v} = 5\) and the angle between \(\bm{u}\) and \(\bm{v}\) is 30°, what must \(|\bm{u}|\) be?
4. What is the angle between the vectors \(\bm{u} = \langle 1,3 \rangle\) and \(\bm{v} = \langle -4,2 \rangle?\)
5. What is the angle between the vectors \(\bm{u} = \langle 5,4 \rangle\) and \(\bm{v} = \langle 1,-3 \rangle?\)
6. Make a sketch of the vectors \(\bm{u} = \langle 1,3 \rangle\) and \(\bm{v} = \langle -4,2 \rangle.\) Then on your sketch add in vectors for the projection of \(\bm{u}\) onto \(\bm{v}\) (\(\operatorname{proj}_{\bm{v}}(\bm{u})\)) and the projection of \(\bm{v}\) onto \(\bm{u}\) (\(\operatorname{proj}_{\bm{u}}(\bm{v})\)), and calculate the components of these vectors explicitly.
7. Make a sketch of the vectors \(\bm{u} = \langle 5,4 \rangle\) and \(\bm{v} = \langle 1,-3 \rangle.\) Then on your sketch add in vectors for the projection of \(\bm{u}\) onto \(\bm{v}\) (\(\operatorname{proj}_{\bm{v}}(\bm{u})\)) and the projection of \(\bm{v}\) onto \(\bm{u}\) (\(\operatorname{proj}_{\bm{u}}(\bm{v})\)), and calculate the components of these vectors explicitly.