Further vectors – further maths worksheet with answers to download as PDF

The vectors \(\mathbf a=2\mathbf i-\mathbf j+3\mathbf k\) and \(\mathbf b=\mathbf i+4\mathbf j-2\mathbf k\) are given.

1. Calculate \(\mathbf a\cdot\mathbf b\), \(|\mathbf a|\) and \(|\mathbf b|\).
2. Find the angle between \(\mathbf a\) and \(\mathbf b\) to 1 decimal place.
3. Show that \(3\mathbf i+\mathbf j-2\mathbf k\) is perpendicular to \(\mathbf i-\mathbf j+\mathbf k\).
4. Find the value of \(k\) for which \(k\mathbf i+2\mathbf j-\mathbf k\) is perpendicular to \(3\mathbf i+k\mathbf j+4\mathbf k\).

Use the same vectors \(\mathbf a=2\mathbf i-\mathbf j+3\mathbf k\) and \(\mathbf b=\mathbf i+4\mathbf j-2\mathbf k\).

1. Calculate \(\mathbf a\times\mathbf b\) using a determinant.
2. Verify that \(\mathbf a\times\mathbf b\) is perpendicular to both \(\mathbf a\) and \(\mathbf b\).
3. Find the area of the parallelogram with sides \(\mathbf a\) and \(\mathbf b\), and of the triangle with these two sides.

A line passes through \(A(1,2,-1)\) and has direction vector \(2\mathbf i-\mathbf j+3\mathbf k\).

1. Write down the vector equation of the line.
2. Write the equation in Cartesian form.
3. A second line passes through \(P(3,0,1)\) and \(Q(5,-2,4)\). Find a vector equation of line \(PQ\).
4. Show that \((7,-4,7)\) lies on \(PQ\) but \((1,2,-1)\) does not.

Find the Cartesian equation of each plane.

1. The plane through \(A(1,0,2)\) with normal vector \(2\mathbf i-\mathbf j+3\mathbf k\). Show that \((3,1,1)\) lies in the plane.
2. The plane through \(A(1,0,0)\), \(B(0,2,0)\) and \(C(0,0,3)\).
3. Write the plane in part (a) in the form \(\mathbf r\cdot\mathbf n=d\).

Find each angle to 1 decimal place.

1. The angle \(\varphi\) between the line \(\mathbf r=\mathbf a+\lambda(\mathbf i+2\mathbf j+2\mathbf k)\) and the plane \(\mathbf r\cdot(2\mathbf i-\mathbf j+2\mathbf k)=7\).
2. The angle between the planes \(x+2y+2z=5\) and \(3x+4z=2\).

The plane \(\Pi\) has equation \(2x-2y+z=5\) and the point \(P\) is \((2,3,-1)\).

1. Find the perpendicular distance from \(P\) to \(\Pi\).
2. Write down the equation of the line through \(P\) perpendicular to \(\Pi\).
3. Find the coordinates of the foot of the perpendicular and verify it lies on \(\Pi\).

A survey drone flies in a straight line. Relative to a mast at the origin \(O\) (distances in metres, \(z\) vertical), its position is \(\mathbf r=\begin{pmatrix}10\\-20\\30\end{pmatrix}+t\begin{pmatrix}2\\4\\-1\end{pmatrix}\), where \(t\) is time in seconds.

1. Find the speed of the drone.
2. Find the value of \(t\) when the drone is closest to the mast.
3. Find the shortest distance from the mast to the drone’s path, to 1 decimal place.
4. The drone must stay at least 30 m from the mast. Does it?

The corners of a triangular roof panel are \(P(0,0,4)\), \(Q(6,0,4)\) and \(R(0,4,6)\), where lengths are in metres and \(z\) is the height.

1. Find the vectors \(\overrightarrow{PQ}\) and \(\overrightarrow{PR}\) and their cross product.
2. Show that the plane of the roof has equation \(2z-y=8\).
3. Find the area of the panel.
4. Find the angle between the roof and the horizontal ground, to 1 decimal place.

A laser beam is fired from the point \(A(0,0,9)\) in the direction \(\mathbf i+2\mathbf j-3\mathbf k\) towards a flat mirror lying in the plane \(2x+y+2z=12\) (lengths in metres).

1. Write the vector equation of the beam.
2. Find the position where the beam meets the mirror.
3. Find the angle between the beam and the mirror, to 1 decimal place.
4. What would happen if the beam were fired in the direction \(\mathbf i+2\mathbf j-2\mathbf k\)?

Two aircraft fly along straight lines (units in km, \(z\) is altitude): \(l_1\): \(\mathbf r=\begin{pmatrix}1\\2\\3\end{pmatrix}+\lambda\begin{pmatrix}1\\-1\\2\end{pmatrix}\) and \(l_2\): \(\mathbf r=\begin{pmatrix}2\\0\\-1\end{pmatrix}+\mu\begin{pmatrix}2\\1\\1\end{pmatrix}\).

1. Show that the lines do not intersect.
2. Find a vector perpendicular to both directions.
3. Find the shortest distance between the paths to 2 decimal places.
4. Air traffic control requires paths to be at least 3 km apart. Is this requirement met?

A crane applies a constant force \(\mathbf F=4\mathbf i-2\mathbf j+5\mathbf k\) newtons to a load that moves through the displacement \(\mathbf s=3\mathbf i+\mathbf j+2\mathbf k\) metres.

1. Calculate the work done, \(\mathbf F\cdot\mathbf s\).
2. Find the magnitudes \(|\mathbf F|\) and \(|\mathbf s|\).
3. Find the angle between the force and the displacement to the nearest degree.
4. Find the component of the force in the direction of the displacement.

A spanner applies a force \(\mathbf F=(0,30,-10)\) N at the point with position vector \(\mathbf r=(0.4,0,0.1)\) m measured from the centre of a bolt. The moment (torque) is \(\mathbf r\times\mathbf F\).

1. Calculate the moment vector.
2. Find its magnitude in N m.
3. Find the component of the moment about the \(z\)-axis, which is the axis of the bolt.
4. Explain why a longer spanner would make the bolt easier to turn.

A tent is a pyramid with its apex at the origin \(O\) and base vertices \(A(2,1,0)\), \(B(-1,3,2)\) and \(C(0,-2,4)\) (lengths in metres).

1. Find \(\overrightarrow{OB}\times\overrightarrow{OC}\).
2. The volume of the tetrahedron \(OABC\) is \(\tfrac16\left|\overrightarrow{OA}\cdot\left(\overrightarrow{OB}\times\overrightarrow{OC}\right)\right|\). Find it.
3. Find the area of the triangle \(OBC\).
4. Hence find the perpendicular distance from \(A\) to the plane \(OBC\).

Two submarines move at constant velocities. Submarine A has position \(\mathbf r_A=(1,2,3)+t(2,-1,1)\) and submarine B has position \(\mathbf r_B=(4,-1,2)+t(-1,2,2)\), in kilometres, where \(t\) is the time in hours and \(z\) is height above the sea floor.

1. Show that the paths cross at the point \((3,1,4)\) and state the value of the parameter for each path.
2. Do the submarines collide? Explain.
3. Find the acute angle between the two paths, to 1 decimal place.
4. Find the speed of each submarine.

The line \(l\) is \(\mathbf r=(1,0,2)+\lambda(1,1,1)\) and the plane \(\Pi\) is \(x+2y-z=4\).

1. Find the coordinates of the point where \(l\) meets \(\Pi\).
2. Find the acute angle between \(l\) and \(\Pi\).
3. Show that the point \(P(1,0,2)\) does not lie on \(\Pi\) and find its distance from \(\Pi\).
4. Find the image of \(P\) when it is reflected in \(\Pi\).

The lines \(l_1\): \(\mathbf r=(1,1,0)+\lambda(1,2,1)\) and \(l_2\): \(\mathbf r=(2,-1,3)+\mu(1,2,1)\) are parallel.

1. Find a vector joining a point on \(l_1\) to a point on \(l_2\).
2. Find the Cartesian equation of the plane containing both lines.
3. Find the shortest distance between the lines, in exact form.
4. Explain why the distance does not depend on which points you choose.

Use the lines \(l_1\): \(\mathbf r=(1,2,3)+\lambda(1,-1,2)\) and \(l_2\): \(\mathbf r=(2,0,-1)+\mu(2,1,1)\) from Question 10.

1. Write the vector \(\overrightarrow{AB}\) from \(A=(1+\lambda,2-\lambda,3+2\lambda)\) on \(l_1\) to \(B=(2+2\mu,\mu,-1+\mu)\) on \(l_2\).
2. Use \(\overrightarrow{AB}\cdot\mathbf d_1=0\) and \(\overrightarrow{AB}\cdot\mathbf d_2=0\) to find \(\lambda\) and \(\mu\).
3. Find the closest points \(A\) and \(B\) and the distance \(|AB|\).
4. Verify the distance agrees with Question 10.

The points \(A(2,0,0)\), \(B(0,3,0)\) and \(C(0,0,6)\) and the origin \(O\) form a tetrahedron.

1. Find the Cartesian equation of the plane \(ABC\).
2. Find the volume of the tetrahedron.
3. Find the area of the face \(ABC\).
4. Hence find the perpendicular distance from \(O\) to the plane \(ABC\), and confirm it with the point-to-plane formula.

The plane \(\Pi\) has vector equation \(\mathbf r=\begin{pmatrix}1\\0\\2\end{pmatrix}+\lambda\begin{pmatrix}1\\1\\0\end{pmatrix}+\mu\begin{pmatrix}0\\1\\1\end{pmatrix}\).

1. Find a normal vector to \(\Pi\).
2. Find the Cartesian equation of \(\Pi\).
3. Show that \((3,1,1)\) lies in \(\Pi\) and find the values of \(\lambda\) and \(\mu\).
4. Find the distance of the origin from \(\Pi\).

Let \(\mathbf a\) and \(\mathbf b\) be vectors and let \(\theta\) be the angle between them.

1. Show that \(|\mathbf a\times\mathbf b|=|\mathbf a||\mathbf b|\sin\theta\) using \(\sin^2\theta=1-\cos^2\theta\) and the identity \(|\mathbf a\times\mathbf b|^2=|\mathbf a|^2|\mathbf b|^2-(\mathbf a\cdot\mathbf b)^2\).
2. Verify the identity for \(\mathbf a=(2,-1,3)\) and \(\mathbf b=(1,4,-2)\).
3. Prove the identity for \(\mathbf a=(a_1,a_2,a_3)\), \(\mathbf b=(b_1,b_2,b_3)\) by expanding both sides.
4. Explain why \(|\mathbf a\times\mathbf b|\) is the area of a parallelogram.