Scalar multiplication and parallel vectors Problem set A

Given \(\mathbf{a}=\begin{pmatrix}2\\-3\\4\end{pmatrix}\) and \(\mathbf{b}=\begin{pmatrix}5\\1\\-2\end{pmatrix}\), find:

1. (a) \(3\mathbf{a}+2\mathbf{b}\)
2. (b) \(2\mathbf{a}-3\mathbf{b}\)

Given \(\mathbf{d}=\begin{pmatrix}1\\4\end{pmatrix}\), \(\mathbf{e}=\begin{pmatrix}-3\\2\end{pmatrix}\) and \(\mathbf{f}=\begin{pmatrix}0\\5\end{pmatrix}\), find:

1. (a) \(\mathbf{d}+2\mathbf{e}-3\mathbf{f}\)
2. (b) \(4\mathbf{d}-2\mathbf{e}+\mathbf{f}\)

Given \(\mathbf{a}=\begin{pmatrix}2\\-5\\3\end{pmatrix}\) and \(\mathbf{c}=\begin{pmatrix}8\\p\\q\end{pmatrix}\), find the values of \(p\) and \(q\) so that \(\mathbf{c}\) is parallel to \(\mathbf{a}\).

Given \(\mathbf{m}=\begin{pmatrix}2\\-3\\7\end{pmatrix}\) and \(\mathbf{n}=\begin{pmatrix}p\\6\\q\end{pmatrix}\), find the values of \(p\) and \(q\) so that \(\mathbf{n}\) is parallel to \(\mathbf{m}\).

ABCD is a parallelogram with \(\overrightarrow{AB}=\mathbf{a}\) and \(\overrightarrow{AD}=\mathbf{b}\). M is the midpoint of AD. N is on the line BC extended beyond C, with CN = 2BC. Write these in terms of \(\mathbf{a}\) and \(\mathbf{b}\).

1. (a) \(\overrightarrow{AM}\)
2. (b) \(\overrightarrow{BN}\)
3. (c) \(\overrightarrow{MN}\)

ABCD is a parallelogram with \(\overrightarrow{AB}=\mathbf{a}\) and \(\overrightarrow{AD}=\mathbf{b}\). X is on the line AB extended beyond B, with BX = AB. Y is the midpoint of CD. Write these in terms of \(\mathbf{a}\) and \(\mathbf{b}\).

1. (a) \(\overrightarrow{DY}\)
2. (b) \(\overrightarrow{XC}\)
3. (c) \(\overrightarrow{XY}\)

Two drones have velocities \(\mathbf{u}=\begin{pmatrix}5\\-2\\3\end{pmatrix}\) m/s and \(\mathbf{v}=\begin{pmatrix}-1\\4\\2\end{pmatrix}\) m/s.

1. (a) The velocity of a third drone is \(2\mathbf{u}-3\mathbf{v}\). Find it as a column vector.
2. (b) A fourth drone has velocity \(\mathbf{w}=\begin{pmatrix}-10\\p\\q\end{pmatrix}\) m/s and flies parallel to the first drone. Find \(p\) and \(q\).

The vectors \(\begin{pmatrix}k\\4\end{pmatrix}\) and \(\begin{pmatrix}9\\k\end{pmatrix}\) are parallel. Find the two possible values of \(k\).

Show that \(\mathbf{m}=\begin{pmatrix}6\\-9\\15\end{pmatrix}\) and \(\mathbf{n}=\begin{pmatrix}-4\\6\\-10\end{pmatrix}\) are parallel, and state whether they point the same way.