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

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

1. (a) \(\mathbf{a}=\begin{pmatrix}4 \\ -1 \\ 3\end{pmatrix},\ \mathbf{b}=\begin{pmatrix}12 \\ p \\ q\end{pmatrix};\)
2. (b) \(\mathbf{a}=p\mathbf{i}+4\mathbf{j}-6\mathbf{k},\ \mathbf{b}=3\mathbf{i}+q\mathbf{j}-9\mathbf{k};\)

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

1. (a) find the unit vector in the same direction as \(\mathbf{a}\);
2. (b) find the vector of magnitude 20 in the direction of \(\mathbf{d}\), in column form.

Points A, B and C have position vectors \(\mathbf{a}=\begin{pmatrix}4 \\ -3 \\ 1\end{pmatrix}\), \(\mathbf{b}=\begin{pmatrix}6 \\ 0 \\ -2\end{pmatrix}\) and \(\mathbf{c}=\begin{pmatrix}-1 \\ 2 \\ 5\end{pmatrix}\). 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) \(\mathbf{a}=4\mathbf{i}-\mathbf{j}+3\mathbf{k},\ \mathbf{b}=\mathbf{i}+2\mathbf{j}-\mathbf{k};\)
2. (b) \(\mathbf{a}=\begin{pmatrix}2 \\ 5 \\ -1\end{pmatrix},\ \mathbf{b}=\begin{pmatrix}8 \\ -3 \\ -1\end{pmatrix}.\)

The resultant of \(\mathbf{u}=\begin{pmatrix}8 \\ y \\ -1\end{pmatrix},\ \mathbf{v}=\begin{pmatrix}-5 \\ 4 \\ z\end{pmatrix}\text{ and }\mathbf{w}=\begin{pmatrix}x \\ -7 \\ 4\end{pmatrix}\) is \(\begin{pmatrix}2 \\ -1 \\ 10\end{pmatrix}\). Find \(x\), \(y\) and \(z\).

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

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

In each part, find the unknown.

1. (a) Given \(\mathbf{p}=\begin{pmatrix}3 \\ -2 \\ 1\end{pmatrix}\text{ and }\mathbf{q}=\begin{pmatrix}9 \\ -10 \\ 11\end{pmatrix}\), find the vector \(\mathbf{x}\) such that \(2\mathbf{p}+3\mathbf{x}=\mathbf{q}\).
2. (b) Given \(\mathbf{a}=2\mathbf{i}-\mathbf{j}+3\mathbf{k},\ \mathbf{b}=\mathbf{i}+2\mathbf{j}-\mathbf{k}\text{ and }\mathbf{c}=-5\mathbf{j}+5\mathbf{k}\), find the scalar \(t\) such that \(\mathbf{a}+t\mathbf{b}=\mathbf{c}\).

The vector \((4+2t)\mathbf{i}+(-2-t)\mathbf{j}+(1-3t)\mathbf{k}\) has magnitude 7. Find the possible values of \(t\).

Three forces act on a boat: \(\mathbf{F}_1=\begin{pmatrix}8 \\ -3 \\ 5\end{pmatrix},\ \mathbf{F}_2=\begin{pmatrix}-2 \\ 7 \\ -9\end{pmatrix}\text{ and }\mathbf{F}_3=\begin{pmatrix}4 \\ -1 \\ 3\end{pmatrix}\), in newtons. Find the magnitude of the resultant force, to 3 significant figures.

The diagram shows a rectangular tile PQRS. M is the midpoint of QR, N is the midpoint of RS and T is the point on PQ extended beyond Q such that \(QT=\frac{1}{2}PQ\). Let \(\mathbf{a}=\overrightarrow{PQ}\) and \(\mathbf{b}=\overrightarrow{PS}\). Express in terms of \(\mathbf{a}\) and \(\mathbf{b}\):

1. (a) \(\overrightarrow{PM}\)
2. (b) \(\overrightarrow{MN}\)
3. (c) \(\overrightarrow{ST}\)

In triangle OAB, \(\overrightarrow{OA}=\mathbf{a}\) and \(\overrightarrow{OB}=\mathbf{b}\). M is the midpoint of OB. P is the point on AB such that \(AP:PB=2:3\). Express in terms of \(\mathbf{a}\) and \(\mathbf{b}\):

1. (a) \(\overrightarrow{AB}\)
2. (b) \(\overrightarrow{OP}\)
3. (c) \(\overrightarrow{MP}\)

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

1. (a) Given \(\mathbf{a}=\begin{pmatrix}3 \\ -1 \\ 2\end{pmatrix}\text{ and }\mathbf{b}=\begin{pmatrix}1 \\ 2 \\ -1\end{pmatrix}\), find \(p\) such that \(\mathbf{a}+p\mathbf{b}\) is parallel to \(\begin{pmatrix}-1 \\ 5 \\ -4\end{pmatrix}\).
2. (b) Given \(\mathbf{x}=\begin{pmatrix}3 \\ 5 \\ -2\end{pmatrix}\text{ and }\mathbf{y}=\begin{pmatrix}-6 \\ 1 \\ 4\end{pmatrix}\), find \(\lambda\) and \(k\) such that \(\lambda\mathbf{x}+\mathbf{y}=k\mathbf{j}\).

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

1. (a) Find the values of \(\lambda\) such that \(|\mathbf{a}+\lambda\mathbf{b}|=6\), where \(\mathbf{a}=\begin{pmatrix}1 \\ -3 \\ 2\end{pmatrix}\) and \(\mathbf{b}=\begin{pmatrix}2 \\ 1 \\ -2\end{pmatrix}\).
2. (b) Points A and B are such that \(\overrightarrow{OA}=\begin{pmatrix}-2 \\ 3 \\ 5\end{pmatrix}\) and \(\overrightarrow{OB}=\begin{pmatrix}1 \\ 0 \\ 2\end{pmatrix}+t\begin{pmatrix}2 \\ -1 \\ 1\end{pmatrix}\), where O is the origin. Find the values of \(t\) such that \(AB=6\).
3. (c) Find the smallest possible magnitude of the vector \((2+t)\mathbf{i}+(-1+2t)\mathbf{j}+(2-t)\mathbf{k}\), where \(t\) is a real number.

Points P and Q have position vectors \(\mathbf{p}=4\mathbf{i}-\mathbf{j}+6\mathbf{k}\) and \(\mathbf{q}=-2\mathbf{i}+5\mathbf{j}+2\mathbf{k}\).

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.