3.8. Vectors Exercises#
Answer the following exercises based on the content from this chapter. The solutions can be found in the appendices.
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) the vector pointing from \(U\) to \(V\)
(g) the vector pointing from \(U\) to \(W\)
(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 \(\vec{i}\), \(\vec{j}\) and \(\vec{k}\)
(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?