Mixed practice: scalar and vector products, lines and distances Problem set A

Find the angle between the two vectors.

1. (a) \(\overrightarrow{OA}=\begin{pmatrix}3\\1\\2\end{pmatrix}\) and \(\overrightarrow{OB}=\begin{pmatrix}0\\5\\-1\end{pmatrix}\). Find the angle between \(\overrightarrow{AB}\) and \(\overrightarrow{OA}\).
2. (b) \(P(1, 2, 0)\), \(Q(3, -1, 2)\), \(R(5, 0, 3)\), \(S(2, 4, 1)\). Find the angle between \(\overrightarrow{PR}\) and \(\overrightarrow{QS}\).

Use the scalar product and the vector product to find each angle or magnitude.

1. (a) \(|\mathbf{a}|=7\), \(|\mathbf{b}|=5\) and \(\mathbf{a}\cdot\mathbf{b}=21\). Find the angle between \(\mathbf{a}\) and \(\mathbf{b}\).
2. (b) \(|\mathbf{c}|=8\), \(|\mathbf{d}|=10\) and \(\mathbf{c}\cdot\mathbf{d}=-30\). Find the angle between \(\mathbf{c}\) and \(\mathbf{d}\).
3. (c) \(|\mathbf{a}|=6\), \(\mathbf{a}\cdot\mathbf{b}=15\) and the angle between \(\mathbf{a}\) and \(\mathbf{b}\) is \(40^\circ\). Find \(|\mathbf{b}|\).
4. (d) \(|\mathbf{a}|=6\), \(|\mathbf{b}|=4\) and the angle between them is \(60^\circ\). Find the exact value of \(|\mathbf{a}\times\mathbf{b}|\).
5. (e) \(|\mathbf{a}|=3\), \(|\mathbf{b}|=4\) and \(|\mathbf{a}\times\mathbf{b}|=9\). Find, in radians, the acute angle between \(\mathbf{a}\) and \(\mathbf{b}\).
6. (f) \(|\mathbf{a}|=4\), \(|\mathbf{b}|=3\) and \(\mathbf{a}\times\mathbf{b}=2\mathbf{i} - 3\mathbf{j} - 6\mathbf{k}\). Find, in radians, the acute angle between \(\mathbf{a}\) and \(\mathbf{b}\).

Work with these lines.

1. (a) Find the acute angle between the lines \(\mathbf{r}=\begin{pmatrix}2\\-1\\0\end{pmatrix}+t\begin{pmatrix}1\\2\\2\end{pmatrix}\) and \(\mathbf{r}=\begin{pmatrix}0\\3\\1\end{pmatrix}+s\begin{pmatrix}2\\-1\\-2\end{pmatrix}\).
2. (b) Find the acute angle between the lines \(x=1+2\lambda , y=3-\lambda , z=4\lambda \) and \(x=2-\mu , y=1+3\mu , z=-5+2\mu \).
3. (c) Show that the lines \(\mathbf{r}=\begin{pmatrix}1\\0\\2\end{pmatrix}+\lambda \begin{pmatrix}2\\3\\-1\end{pmatrix}\) and \(\mathbf{r}=\begin{pmatrix}4\\-1\\0\end{pmatrix}+\mu \begin{pmatrix}1\\-1\\-1\end{pmatrix}\) are perpendicular.
4. (d) A line has parametric equations \(x=2+3\lambda , y=5, z=1-4\lambda \). Write down a vector equation of the line, and find the acute angle it makes with the x-axis.

Work with perpendicular vectors.

1. (a) Find a vector that is perpendicular to both \(\mathbf{a}=2\mathbf{i} - \mathbf{j} + 3\mathbf{k}\) and \(\mathbf{b}=\mathbf{i} + 2\mathbf{j} - \mathbf{k}\).
2. (b) Find a unit vector that is perpendicular to both \(\begin{pmatrix}2\\0\\1\end{pmatrix}\) and \(\begin{pmatrix}1\\3\\-1\end{pmatrix}\).
3. (c) Given \(\mathbf{p}=\begin{pmatrix}1\\2\\-3\end{pmatrix}\) and \(\mathbf{q}=\begin{pmatrix}2\\-1\\1\end{pmatrix}\), find the value of \(t\) for which \(\mathbf{p}+t\mathbf{q}\) is perpendicular to \(\begin{pmatrix}1\\3\\2\end{pmatrix}\).
4. (d) The vectors \(t\mathbf{i} - \mathbf{j} + 3\mathbf{k}\) and \(2 t\mathbf{i} + t\mathbf{j} - \mathbf{k}\) are perpendicular. Find the possible values of \(t\).

Properties of the scalar product and the vector product.

1. (a) Explain why \((\mathbf{a}\times\mathbf{b})\cdot\mathbf{b}=0\).
2. (b) Evaluate \((\mathbf{a}\times\mathbf{b})\cdot(3\mathbf{a}-2\mathbf{b})\).
3. (c) Take \(\mathbf{a}=\begin{pmatrix}2\\1\\3\end{pmatrix}\) and \(\mathbf{b}=\begin{pmatrix}1\\-2\\2\end{pmatrix}\). Show that \(|\mathbf{a}\times\mathbf{b}|^2+(\mathbf{a}\cdot\mathbf{b})^2=|\mathbf{a}|^2|\mathbf{b}|^2\).
4. (d) The points are \(A(2, 0, 1)\), \(B(3, 2, 4)\) and \(C(5, 1, 2)\). Calculate \(\mathbf{p}=\overrightarrow{AB}\times\overrightarrow{AC}\) and \(\mathbf{q}=\overrightarrow{BA}\times\overrightarrow{BC}\). What do you notice about \(\mathbf{p}\) and \(\mathbf{q}\)?

The triangle ABC has \(A(1, 2, 3)\), \(B(4, 0, 5)\), \(C(3, 5, -1)\).

1. (a) Find the three angles of the triangle, in degrees.
2. (b) Find the area of the triangle.

The triangle ABC has \(A(1, 0, 2)\), \(B(3, 1, 0)\), \(C(3, 2, 5)\).

1. (a) Show that the angle at A is a right angle.
2. (b) Find the other two angles, in degrees.
3. (c) Find the area of the triangle.

ABCD is a parallelogram. Three of its vertices are \(A(0, 2, 1)\), \(B(3, 3, -1)\), \(C(5, 6, 2)\).

1. (a) Find the coordinates of D.
2. (b) Find the area of the parallelogram.

ABCD has \(A(1, 0, 2)\), \(B(3, 1, 4)\) and \(D(3, 2, 1)\), and \(\overrightarrow{AB}+\overrightarrow{AD}=\overrightarrow{AC}\).

1. (a) Find the coordinates of C.
2. (b) Show that ABCD is a rhombus, by comparing the lengths of \(\overrightarrow{AB}\) and \(\overrightarrow{AD}\).
3. (c) Show that the diagonals AC and BD are perpendicular.

The cuboid ABCDEFGH has \(A(2, 1, 0)\), \(B(7, 1, 0)\), \(D(2, 4, 0)\), \(E(2, 1, 4)\). E is directly above A, F above B, G above C and H above D.

1. (a) Find the coordinates of C, F, G and H.
2. (b) Find the area of the triangle BDE formed by three face diagonals.

A ferry sails in the direction \(\begin{pmatrix}2\\-1\end{pmatrix}\). A river current flows at 15 km/h in the direction \(\begin{pmatrix}4\\3\end{pmatrix}\).

1. (a) Write the velocity of the current as a column vector.
2. (b) Find the component of the current's velocity in the direction the ferry is sailing.

Points A and B have position vectors \(\begin{pmatrix}2\\2\\1\end{pmatrix}\) and \(\begin{pmatrix}2\lambda\\2\lambda\\\lambda\end{pmatrix}\) respectively, and \(C(7, 5, 3)\).

1. (a) Show that B lies on the line OA for every value of \(\lambda\).
2. (b) Find the value of \(\lambda\) for which CB is perpendicular to OA.
3. (c) Hence find the exact distance from C to the line OA.

The line \(l\) has equation \(\mathbf{r}=\begin{pmatrix}1\\-2\\3\end{pmatrix}+\lambda \begin{pmatrix}2\\1\\-2\end{pmatrix}\). The point P has coordinates \((6, -2, -1)\).

1. (a) Find the coordinates of the point M on \(l\) such that PM is perpendicular to \(l\).
2. (b) Show that the point \(Q(-5, -5, 9)\) lies on \(l\).
3. (c) The point R, different from Q, lies on \(l\) and \(|PR|=|PQ|\). Find the coordinates of R.