3.8. Vectors Exercises#
Exercise 3.1
The points \(U\), \(V\) and \(W\) have the following position vectors:
Find:
(a) \(2 \vec{u} + \vec{w}\)
(b) \(\vec{w} - \vec{u}\)
(c) a unit vector pointing in the same direction of \(\vec{u}\)
(d) a unit vector pointing in the opposite direction of \(\vec{v}\)
(e) a vector pointing in the same direction as \(\vec{v}\) but half its length
(f) \(\overrightarrow{UV}\)
(g) \(\overrightarrow{UW}\)
(h) \(\vec{u} \cdot \vec{w}\)
(i) the angle \(\angle VUW\)
(j) show that \(\vec{u}\) is at right angles to \(\vec{v}\)
(k) \(\vec{v} \times \vec{w}\)
Exercise 3.2
Write \(\vec{u} = (2,7,1)^\mathsf{T}\) as:
(a) a linear combination of \(\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}\), \(\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}\) and \(\begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}\).
(b) a linear combination of vectors \(\vec{f}_1 = \begin{pmatrix} 1 \\ -1 \\ 0 \end{pmatrix}, \vec{f}_2 = \begin{pmatrix} 0 \\ 2 \\ 0 \end{pmatrix}\) and \(\vec{f}_3 = \begin{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix}\).
Exercise 3.3
Find \(k\) such that the vectors \(\vec{u}\) and \(\vec{v}\) are perpendicular:
(a) \(\vec{u} = \begin{pmatrix} 1 \\ k \\ -2 \end{pmatrix}\) and \(\vec{v} = \begin{pmatrix} 2 \\ -5 \\ 4 \end{pmatrix}\) in \(\mathbb{R}^3\).
(b) \(\vec{u} = \begin{pmatrix} 1 \\ 0 \\ k + 2 \\ -1 \\ 2 \end{pmatrix}\) and \(\vec{v} = \begin{pmatrix} 1 \\ k \\ -2 \\ 1 \\ 2 \end{pmatrix}\) in \(\mathbb{R}^5\).
Exercise 3.4
Which pair of the following vectors is perpendicular? For the remaining pairs, what is the angle between them?