Mixed practice: parallel vectors, unit vectors, displacement, distance and parameters Problem set B

In each part the vectors \(\mathbf{a}\) and \(\mathbf{b}\) are parallel. Find \(p\) and \(q\).

1. (a) \(\mathbf{a}=\begin{pmatrix}-2 \\ 3 \\ 1\end{pmatrix},\ \mathbf{b}=\begin{pmatrix}8 \\ p \\ q\end{pmatrix};\)
2. (b) \(\mathbf{a}=p\mathbf{i}-9\mathbf{j}+3\mathbf{k},\ \mathbf{b}=4\mathbf{i}+q\mathbf{j}+\mathbf{k};\)

Given \(\mathbf{a}=\begin{pmatrix}2 \\ -3 \\ 6\end{pmatrix}\) and \(\mathbf{d}=4\mathbf{i}-4\mathbf{j}+7\mathbf{k}\):

1. (a) find the unit vector in the same direction as \(\mathbf{a}\);
2. (b) find the vector of magnitude 15 in the direction of \(\mathbf{d}\), in column form.

Points A, B and C have position vectors \(\mathbf{a}=\begin{pmatrix}-2 \\ 5 \\ 3\end{pmatrix}\), \(\mathbf{b}=\begin{pmatrix}1 \\ -4 \\ 6\end{pmatrix}\) and \(\mathbf{c}=\begin{pmatrix}3 \\ 1 \\ -5\end{pmatrix}\). Find:

1. (a) \(\overrightarrow{AB}\);
2. (b) \(\overrightarrow{AC}\);
3. (c) \(\overrightarrow{CB}\).

Find the exact distance \(AB\) between the points with position vectors:

1. (a) \(\mathbf{a}=2\mathbf{i}+3\mathbf{j}-5\mathbf{k},\ \mathbf{b}=-2\mathbf{i}+\mathbf{j}-3\mathbf{k};\)
2. (b) \(\mathbf{a}=\begin{pmatrix}-1 \\ 0 \\ 4\end{pmatrix},\ \mathbf{b}=\begin{pmatrix}3 \\ -3 \\ 16\end{pmatrix}.\)

The resultant of \(\mathbf{u}=\begin{pmatrix}x \\ 3 \\ -6\end{pmatrix},\ \mathbf{v}=\begin{pmatrix}5 \\ y \\ 2\end{pmatrix}\text{ and }\mathbf{w}=\begin{pmatrix}-9 \\ -8 \\ z\end{pmatrix}\) is \(\begin{pmatrix}-3 \\ -2 \\ 1\end{pmatrix}\). Find \(x\), \(y\) and \(z\).

Given \(\mathbf{a}=\begin{pmatrix}-1 \\ 2 \\ 3\end{pmatrix},\ \mathbf{b}=\begin{pmatrix}4 \\ 0 \\ -3\end{pmatrix}\text{ and }\mathbf{c}=\begin{pmatrix}2 \\ -5 \\ 1\end{pmatrix}\), find:

1. (a) \(3\mathbf{a}+2\mathbf{b}-2\mathbf{c}\);
2. (b) the exact value of \(|-\mathbf{a}+3\mathbf{b}|\).

In each part, find the unknown.

1. (a) Given \(\mathbf{p}=\begin{pmatrix}2 \\ 0 \\ -1\end{pmatrix}\text{ and }\mathbf{q}=\begin{pmatrix}4 \\ 4 \\ -13\end{pmatrix}\), find the vector \(\mathbf{x}\) such that \(5\mathbf{p}-2\mathbf{x}=\mathbf{q}\).
2. (b) Given \(\mathbf{a}=\mathbf{i}+4\mathbf{j}-3\mathbf{k},\ \mathbf{b}=2\mathbf{i}-\mathbf{j}+\mathbf{k}\text{ and }\mathbf{c}=7\mathbf{i}+\mathbf{j}\), find the scalar \(t\) such that \(\mathbf{a}+t\mathbf{b}=\mathbf{c}\).

The vector \((2-t)\mathbf{i}+(1+t)\mathbf{j}+(-4+2t)\mathbf{k}\) has magnitude 9. Find the possible values of \(t\).

Three tugs pull a ship with the forces \(\mathbf{F}_1=\begin{pmatrix}-6 \\ 4 \\ 2\end{pmatrix},\ \mathbf{F}_2=\begin{pmatrix}9 \\ -8 \\ 5\end{pmatrix}\text{ and }\mathbf{F}_3=\begin{pmatrix}1 \\ 6 \\ -7\end{pmatrix}\), in kilonewtons. Find the magnitude of the resultant force, to 3 significant figures.

The diagram shows a rectangular lawn EFGH. J is the midpoint of FG, K is the midpoint of GH and L is the point on EF extended beyond F such that \(FL=\frac{1}{4}EF\). Let \(\mathbf{a}=\overrightarrow{EF}\) and \(\mathbf{b}=\overrightarrow{EH}\). Express in terms of \(\mathbf{a}\) and \(\mathbf{b}\):

1. (a) \(\overrightarrow{EJ}\)
2. (b) \(\overrightarrow{JK}\)
3. (c) \(\overrightarrow{KL}\)

In triangle OAB, \(\overrightarrow{OA}=\mathbf{a}\) and \(\overrightarrow{OB}=\mathbf{b}\). M is the midpoint of OA. N is the point on AB such that \(BN:NA=1:3\). Express in terms of \(\mathbf{a}\) and \(\mathbf{b}\):

1. (a) \(\overrightarrow{BA}\)
2. (b) \(\overrightarrow{ON}\)
3. (c) \(\overrightarrow{MN}\)

In each part, find the scalar or scalars asked for.

1. (a) Given \(\mathbf{a}=\begin{pmatrix}1 \\ -1 \\ 4\end{pmatrix}\text{ and }\mathbf{b}=\begin{pmatrix}2 \\ 1 \\ -1\end{pmatrix}\), find \(p\) such that \(\mathbf{a}+p\mathbf{b}\) is parallel to \(\begin{pmatrix}14 \\ 4 \\ 2\end{pmatrix}\).
2. (b) Given \(\mathbf{x}=\begin{pmatrix}2 \\ -3 \\ 4\end{pmatrix}\text{ and }\mathbf{y}=\begin{pmatrix}-3 \\ 12 \\ -16\end{pmatrix}\), find \(\lambda\) and \(k\) such that \(\lambda\mathbf{x}+\mathbf{y}=k\mathbf{i}\).

Use your calculator where it helps. Give answers to 3 significant figures.

1. (a) Find the values of \(\lambda\) such that \(|\mathbf{a}+\lambda\mathbf{b}|=5\), where \(\mathbf{a}=\begin{pmatrix}-1 \\ 2 \\ 2\end{pmatrix}\) and \(\mathbf{b}=\begin{pmatrix}3 \\ -1 \\ 1\end{pmatrix}\).
2. (b) Points A and B are such that \(\overrightarrow{OA}=\begin{pmatrix}4 \\ -1 \\ 2\end{pmatrix}\) and \(\overrightarrow{OB}=\begin{pmatrix}0 \\ 3 \\ -1\end{pmatrix}+t\begin{pmatrix}1 \\ 2 \\ 2\end{pmatrix}\), where O is the origin. Find the values of \(t\) such that \(AB=8\).
3. (c) Find the smallest possible magnitude of the vector \((-3+2t)\mathbf{i}+(2+t)\mathbf{j}+(-1+3t)\mathbf{k}\), where \(t\) is a real number.

Points P and Q have position vectors \(\mathbf{p}=-3\mathbf{i}+2\mathbf{j}+5\mathbf{k}\) and \(\mathbf{q}=7\mathbf{i}-4\mathbf{j}-\mathbf{k}\).

1. (a) Find the position vector of the midpoint M of PQ.
2. (b) The point R lies on the line PQ, beyond Q, so that \(QR=QM\). Find the coordinates of R.