Vectors#
Solution to Exercise 3.1
(a) \(2\vec{u} + \vec{w} = 2\begin{pmatrix} 2 \\ 3 \end{pmatrix} + \begin{pmatrix} 1 \\ 6 \end{pmatrix} = \begin{pmatrix} 4 \\ 6 \end{pmatrix} + \begin{pmatrix} 1 \\ 6 \end{pmatrix} = \begin{pmatrix} 5 \\ 12 \end{pmatrix}\)
(b) \(\vec{w} - \vec{u} = \begin{pmatrix} 1 \\ 6 \end{pmatrix} - \begin{pmatrix} 2 \\ 3 \end{pmatrix} = \begin{pmatrix} 1 - 2 \\ 6 - 3 \end{pmatrix} = \begin{pmatrix} -1 \\ 3 \end{pmatrix}\)
(c) \(\hat{\vec{u}} = \dfrac{\vec{u}}{\|\vec{u}\|} = \dfrac{1}{\sqrt{13}} \begin{pmatrix} 2 \\ 3 \end{pmatrix} = \begin{pmatrix} \frac{2}{\sqrt{13}} \\ \frac{3}{\sqrt{13}} \end{pmatrix}\)
(d) \(-\hat{\vec{v}} = -\dfrac{\vec{v}}{\|\vec{v}\|} = -\dfrac{1}{\sqrt{13}} \begin{pmatrix} 3 \\ -2 \end{pmatrix} = \begin{pmatrix} -\frac{3}{\sqrt{13}} \\ \frac{2}{\sqrt{13}} \end{pmatrix}\)
(e) \(\dfrac{1}{2}\vec{v} = \dfrac{1}{2} \begin{pmatrix} 3 \\ -2 \end{pmatrix} = \begin{pmatrix} 3 / 2 \\ -2 / 2 \end{pmatrix}\)
(f) \(\vec{v} - \vec{u} = \begin{pmatrix} 3 \\ -2 \end{pmatrix} - \begin{pmatrix} 2 \\ 3 \end{pmatrix}= \begin{pmatrix} 1 \\ -5 \end{pmatrix}\)
(g) \(\vec{w} - \vec{u} = \begin{pmatrix} 1 \\ 6 \end{pmatrix} - \begin{pmatrix} 2 \\ 3 \end{pmatrix} = \begin{pmatrix} -1 \\ 3 \end{pmatrix}\)
(h) \(\vec{u} \cdot \vec{w} = \begin{pmatrix} 2 \\ 3 \end{pmatrix} \cdot \begin{pmatrix} 1 \\ 6 \end{pmatrix} = 2 \times 1 + 3 \times 6 = 20\)
(i) Using equation (3.4)
(j) \(\vec{u} \cdot \vec{v} = \begin{pmatrix} 2 \\ 3 \end{pmatrix} \cdot \begin{pmatrix} 3 \\ -2 \end{pmatrix} = 2 \times 3 + 3 \times (-2) = 0\)
(k) \(\vec{v} \times \vec{w} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ 3 & -2 & 0 \\ 1 & 6 & 0 \end{vmatrix} = 0\vec{i} - 0 \vec{j} + 20 \vec{k} = \begin{pmatrix} 0 \\ 0 \\ 20 \end{pmatrix}\)
Solution to Exercise 3.2
(a) \(\vec{u} = 2 \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} + 7 \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} + \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} = 2 \vec{i} + 7 \vec{j} + \vec{k}\)
(b)
Therefore \(\vec{u} = 3 \vec{f}_1 + 5 \vec{f}_2 - \vec{f}_3\).
Solution to Exercise 3.3
(a) If \(\vec{u}\) and \(\vec{v}\) are perpendicular then \(\vec{u} \cdot \vec{v} = 0\).
(b)
Solution to Exercise 3.4
Therefore \(\vec{u} \perp \vec{v}\). The angle between \(\vec{u}\) and \(\vec{w}\) is
and the angle between \(\vec{v}\) and \(\vec{w}\) is