Scalar multiplication and parallel vectors Problem set B One question at a time

Given \(\mathbf{p}=\begin{pmatrix}-4\\2\\3\end{pmatrix}\) and \(\mathbf{q}=\begin{pmatrix}1\\-5\\6\end{pmatrix}\), find:

1. (a) \(2\mathbf{p}+5\mathbf{q}\)

   • \(2\mathbf{p}+5\mathbf{q}=2\begin{pmatrix}-4\\2\\3\end{pmatrix}+5\begin{pmatrix}1\\-5\\6\end{pmatrix}\)
   • \(=\begin{pmatrix}-8\\4\\6\end{pmatrix}+\begin{pmatrix}5\\-25\\30\end{pmatrix}\)
   • \(=\begin{pmatrix}-3\\-21\\36\end{pmatrix}\)

2. (b) \(3\mathbf{p}-4\mathbf{q}\)

   • \(3\mathbf{p}-4\mathbf{q}=3\begin{pmatrix}-4\\2\\3\end{pmatrix}-4\begin{pmatrix}1\\-5\\6\end{pmatrix}\)
   • \(=\begin{pmatrix}-12\\6\\9\end{pmatrix}-\begin{pmatrix}4\\-20\\24\end{pmatrix}\)
   • \(=\begin{pmatrix}-16\\26\\-15\end{pmatrix}\)

Given \(\mathbf{x}=\begin{pmatrix}6\\-2\end{pmatrix}\), \(\mathbf{y}=\begin{pmatrix}2\\7\end{pmatrix}\) and \(\mathbf{z}=\begin{pmatrix}-4\\0\end{pmatrix}\), find:

1. (a) \(2\mathbf{x}+\mathbf{y}-2\mathbf{z}\)

   • \(2\mathbf{x}+\mathbf{y}-2\mathbf{z}=2\begin{pmatrix}6\\-2\end{pmatrix}+\begin{pmatrix}2\\7\end{pmatrix}-2\begin{pmatrix}-4\\0\end{pmatrix}\)
   • \(=\begin{pmatrix}12\\-4\end{pmatrix}+\begin{pmatrix}2\\7\end{pmatrix}-\begin{pmatrix}-8\\0\end{pmatrix}\)
   • \(=\begin{pmatrix}22\\3\end{pmatrix}\)

2. (b) \(5\mathbf{x}-3\mathbf{y}+4\mathbf{z}\)

   • \(5\mathbf{x}-3\mathbf{y}+4\mathbf{z}=5\begin{pmatrix}6\\-2\end{pmatrix}-3\begin{pmatrix}2\\7\end{pmatrix}+4\begin{pmatrix}-4\\0\end{pmatrix}\)
   • \(=\begin{pmatrix}30\\-10\end{pmatrix}-\begin{pmatrix}6\\21\end{pmatrix}+\begin{pmatrix}-16\\0\end{pmatrix}\)
   • \(=\begin{pmatrix}8\\-31\end{pmatrix}\)

Given \(\mathbf{u}=\begin{pmatrix}3\\-4\\9\end{pmatrix}\) and \(\mathbf{v}=\begin{pmatrix}15\\p\\q\end{pmatrix}\), find the values of \(p\) and \(q\) so that \(\mathbf{v}\) is parallel to \(\mathbf{u}\).

• \(\mathbf{v}=t\mathbf{u}\)
• Equal components: \(15=3t\), so \(t=5\)
• Use t for the other two components: \(p=-4t=-20\) and \(q=9t=45\)

Given \(\mathbf{r}=\begin{pmatrix}4\\-6\\10\end{pmatrix}\) and \(\mathbf{s}=\begin{pmatrix}p\\9\\q\end{pmatrix}\), find the values of \(p\) and \(q\) so that \(\mathbf{s}\) is parallel to \(\mathbf{r}\).

• \(\mathbf{s}=t\mathbf{r}\)
• Equal components: \(9=-6t\), so \(t=-\frac{3}{2}\)
• Use t for the other two components: \(p=4t=-6\) and \(q=10t=-15\)

PQRS is a parallelogram with \(\overrightarrow{PQ}=\mathbf{u}\) and \(\overrightarrow{PS}=\mathbf{v}\). K is the midpoint of PQ. L is on the line SR extended beyond R, with RL = 2SR. Write these in terms of \(\mathbf{u}\) and \(\mathbf{v}\).

1. (a) \(\overrightarrow{PK}\)

   • \(\overrightarrow{PK}=\frac{1}{2}\mathbf{u}\)

2. (b) \(\overrightarrow{SL}\)

   • \(\overrightarrow{SL}=\overrightarrow{SR}+\overrightarrow{RL}\)
   • \(=\mathbf{u}+2\mathbf{u}\)
   • \(=3\mathbf{u}\)

3. (c) \(\overrightarrow{KL}\)

   • \(\overrightarrow{KL}=\overrightarrow{KQ}+\overrightarrow{QR}+\overrightarrow{RL}\)
   • \(=\frac{1}{2}\mathbf{u}+\mathbf{v}+2\mathbf{u}\)
   • \(=\frac{5}{2}\mathbf{u}+\mathbf{v}\)

PQRS is a parallelogram with \(\overrightarrow{PQ}=\mathbf{u}\) and \(\overrightarrow{PS}=\mathbf{v}\). X is on the line PS extended beyond S, with SX = \(\frac{1}{2}\)PS. Y is the midpoint of QR. Write these in terms of \(\mathbf{u}\) and \(\mathbf{v}\).

1. (a) \(\overrightarrow{QY}\)

   • \(\overrightarrow{QY}=\frac{1}{2}\mathbf{v}\)

2. (b) \(\overrightarrow{XR}\)

   • \(\overrightarrow{XR}=\overrightarrow{XS}+\overrightarrow{SR}\)
   • \(=-\frac{1}{2}\mathbf{v}+\mathbf{u}\)
   • \(=\mathbf{u}-\frac{1}{2}\mathbf{v}\)

3. (c) \(\overrightarrow{XY}\)

   • \(\overrightarrow{XY}=\overrightarrow{XR}+\overrightarrow{RY}\)
   • \(=-\frac{1}{2}\mathbf{v}+\mathbf{u}-\frac{1}{2}\mathbf{v}\)
   • \(=\mathbf{u}-\mathbf{v}\)

Two forces on a crate are \(\mathbf{f}=\begin{pmatrix}3\\-4\\6\end{pmatrix}\) N and \(\mathbf{g}=\begin{pmatrix}2\\5\\-1\end{pmatrix}\) N.

1. (a) A third force is \(4\mathbf{f}-\mathbf{g}\). Find it as a column vector.

   • \(4\mathbf{f}-\mathbf{g}=4\begin{pmatrix}3\\-4\\6\end{pmatrix}-\begin{pmatrix}2\\5\\-1\end{pmatrix}\)
   • \(=\begin{pmatrix}12\\-16\\24\end{pmatrix}-\begin{pmatrix}2\\5\\-1\end{pmatrix}\)
   • \(=\begin{pmatrix}10\\-21\\25\end{pmatrix}\)

2. (b) A fourth force \(\mathbf{w}=\begin{pmatrix}9\\p\\q\end{pmatrix}\) N acts along the same line as the first force. Find \(p\) and \(q\).

   • \(\mathbf{w}=t\mathbf{f}\)
   • First components: \(9=3t\), so \(t=3\)
   • \(p=-4t=-12\) and \(q=6t=18\)

The vectors \(\begin{pmatrix}k\\5\end{pmatrix}\) and \(\begin{pmatrix}20\\k\end{pmatrix}\) are parallel. Find the two possible values of \(k\).

• Parallel: \(\begin{pmatrix}20\\k\end{pmatrix}=t\begin{pmatrix}k\\5\end{pmatrix}\)
• Second components: \(k=5t\), so \(t=\frac{k}{5}\)
• First components: \(20=tk=\frac{k^2}{5}\), so \(k^2=100\)
• \(k=10\) or \(k=-10\)

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

• Find \(t\) from the first components: \(t=15\div(-10)=-\frac{3}{2}\)
• Check the other components: \(-\frac{3}{2}\times4=-6\), \(-\frac{3}{2}\times6=-9\)
• Every component fits, so \(\mathbf{v}=-\frac{3}{2}\mathbf{u}\)
• t is negative, so they point opposite ways