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

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

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

   • \(3\mathbf{a}+2\mathbf{b}=3\begin{pmatrix}2\\-3\\4\end{pmatrix}+2\begin{pmatrix}5\\1\\-2\end{pmatrix}\)
   • \(=\begin{pmatrix}6\\-9\\12\end{pmatrix}+\begin{pmatrix}10\\2\\-4\end{pmatrix}\)
   • \(=\begin{pmatrix}16\\-7\\8\end{pmatrix}\)

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

   • \(2\mathbf{a}-3\mathbf{b}=2\begin{pmatrix}2\\-3\\4\end{pmatrix}-3\begin{pmatrix}5\\1\\-2\end{pmatrix}\)
   • \(=\begin{pmatrix}4\\-6\\8\end{pmatrix}-\begin{pmatrix}15\\3\\-6\end{pmatrix}\)
   • \(=\begin{pmatrix}-11\\-9\\14\end{pmatrix}\)

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) \(\mathbf{d}+2\mathbf{e}-3\mathbf{f}\)

   • \(\mathbf{d}+2\mathbf{e}-3\mathbf{f}=\begin{pmatrix}1\\4\end{pmatrix}+2\begin{pmatrix}-3\\2\end{pmatrix}-3\begin{pmatrix}0\\5\end{pmatrix}\)
   • \(=\begin{pmatrix}1\\4\end{pmatrix}+\begin{pmatrix}-6\\4\end{pmatrix}-\begin{pmatrix}0\\15\end{pmatrix}\)
   • \(=\begin{pmatrix}-5\\-7\end{pmatrix}\)

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

   • \(4\mathbf{d}-2\mathbf{e}+\mathbf{f}=4\begin{pmatrix}1\\4\end{pmatrix}-2\begin{pmatrix}-3\\2\end{pmatrix}+\begin{pmatrix}0\\5\end{pmatrix}\)
   • \(=\begin{pmatrix}4\\16\end{pmatrix}-\begin{pmatrix}-6\\4\end{pmatrix}+\begin{pmatrix}0\\5\end{pmatrix}\)
   • \(=\begin{pmatrix}10\\17\end{pmatrix}\)

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 \(\mathbf{a}\).

• \(\mathbf{c}=t\mathbf{a}\)
• Equal components: \(8=2t\), so \(t=4\)
• Use t for the other two components: \(p=-5t=-20\) and \(q=3t=12\)

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 \(\mathbf{m}\).

• \(\mathbf{n}=t\mathbf{m}\)
• Equal components: \(6=-3t\), so \(t=-2\)
• Use t for the other two components: \(p=2t=-4\) and \(q=7t=-14\)

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) \(\overrightarrow{AM}\)

   • \(\overrightarrow{AM}=\frac{1}{2}\mathbf{b}\)

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

   • \(\overrightarrow{BN}=\overrightarrow{BC}+\overrightarrow{CN}\)
   • \(=\mathbf{b}+2\mathbf{b}\)
   • \(=3\mathbf{b}\)

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

   • \(\overrightarrow{MN}=\overrightarrow{MD}+\overrightarrow{DC}+\overrightarrow{CN}\)
   • \(=\frac{1}{2}\mathbf{b}+\mathbf{a}+2\mathbf{b}\)
   • \(=\mathbf{a}+\frac{5}{2}\mathbf{b}\)

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) \(\overrightarrow{DY}\)

   • \(\overrightarrow{DY}=\frac{1}{2}\mathbf{a}\)

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

   • \(\overrightarrow{XC}=\overrightarrow{XB}+\overrightarrow{BC}\)
   • \(=-\mathbf{a}+\mathbf{b}\)

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

   • \(\overrightarrow{XY}=\overrightarrow{XC}+\overrightarrow{CY}\)
   • \(=-\mathbf{a}+\mathbf{b}-\frac{1}{2}\mathbf{a}\)
   • \(=-\frac{3}{2}\mathbf{a}+\mathbf{b}\)

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.

   • \(2\mathbf{u}-3\mathbf{v}=2\begin{pmatrix}5\\-2\\3\end{pmatrix}-3\begin{pmatrix}-1\\4\\2\end{pmatrix}\)
   • \(=\begin{pmatrix}10\\-4\\6\end{pmatrix}-\begin{pmatrix}-3\\12\\6\end{pmatrix}\)
   • \(=\begin{pmatrix}13\\-16\\0\end{pmatrix}\)

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 \(q\).

   • \(\mathbf{w}=t\mathbf{u}\)
   • First components: \(-10=5t\), so \(t=-2\)
   • \(p=-2t=4\) and \(q=3t=-6\)

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

• Parallel: \(\begin{pmatrix}9\\k\end{pmatrix}=t\begin{pmatrix}k\\4\end{pmatrix}\)
• Second components: \(k=4t\), so \(t=\frac{k}{4}\)
• First components: \(9=tk=\frac{k^2}{4}\), so \(k^2=36\)
• \(k=6\) or \(k=-6\)

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.

• Find \(t\) from the first components: \(t=-4\div6=-\frac{2}{3}\)
• Check the other components: \(-\frac{2}{3}\times(-9)=6\), \(-\frac{2}{3}\times15=-10\)
• Every component fits, so \(\mathbf{n}=-\frac{2}{3}\mathbf{m}\)
• t is negative, so they point opposite ways