Mixed practice: lines in 3D, parametric form and intersections Problem set B

The points \(A(-1, 4, 6)\), \(B(-3, 7, 7)\) and \(C(-7, 13, 11)\) are in three dimensions.

1. (a) Find a vector equation of the line \(AB\).
2. (b) Determine whether \(C\) lies on the line \(AB\).

Write down a vector equation of each line.

1. (a) The line through the point \((0, 5, -2)\) with direction \(\begin{pmatrix} -4 \\ 3 \\ 1 \end{pmatrix}\).
2. (b) The line through the point \((6, 1, -3)\) with direction \(\mathbf{i} + 5\mathbf{j} - 2\mathbf{k}\).
3. (c) The line through the point \((-4, 1)\) with direction \(\begin{pmatrix} 6 \\ -1 \end{pmatrix}\).

Write each line in the form asked.

1. (a) Write \(\mathbf{r} = \begin{pmatrix} -3 \\ 1 \\ 5 \end{pmatrix} + \lambda\begin{pmatrix} 4 \\ -6 \\ 3 \end{pmatrix}\) in parametric form.
2. (b) Write \(\mathbf{r} = (4\mathbf{j} - 2\mathbf{k}) + \lambda(3\mathbf{i} - \mathbf{k})\) in parametric form.
3. (c) A line has parametric equations \(x = -1 + 3\lambda\), \(y = 7\), \(z = \frac{\lambda - 6}{3}\). Write it as a vector equation.

A line \(l\) has equation \(\mathbf{r} = \begin{pmatrix} 1 \\ 1 \\ -4 \end{pmatrix} + \lambda\begin{pmatrix} -1 \\ 4 \\ 3 \end{pmatrix}\). Find parametric equations of the line parallel to \(l\) that passes through the point \((-2, 6, 0)\).

A line has parametric equations \(x = 3 + 2\lambda, y = -1 - \lambda, z = 4 + 3\lambda\).

1. (a) Determine whether the point \((-1, 2, -2)\) lies on the line.
2. (b) The point \((9, p, q)\) lies on the line. Find \(p\) and \(q\).

Find the point of intersection of the lines \(\mathbf{r} = (-2\mathbf{i} + 9\mathbf{k}) + \lambda(2\mathbf{i} - \mathbf{j} + 3\mathbf{k})\) and \(\mathbf{r} = \begin{pmatrix} -7 \\ -5 \\ 12 \end{pmatrix} + \mu\begin{pmatrix} 1 \\ 2 \\ -2 \end{pmatrix}\).

Decide about the lines in each part.

1. (a) Show that the lines \(\mathbf{r} = \begin{pmatrix} -1 \\ 2 \\ 7 \end{pmatrix} + \lambda\begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix}\) and \(\mathbf{r} = \begin{pmatrix} -5 \\ 1 \\ 7 \end{pmatrix} + \mu\begin{pmatrix} 3 \\ -1 \\ 1 \end{pmatrix}\) do not intersect.
2. (b) The lines \(l_1\) and \(l_2\) have parametric equations \(l_1: x = 3 - \lambda, y = 1 + 2\lambda, z = -9 + 3\lambda\) and \(l_2: x = 3 + 2\mu, y = 6 + \mu, z = -5 - 2\mu\). Show that they intersect and find the point of intersection.

The line \(l\) has equation \(\mathbf{r} = \begin{pmatrix} -2 \\ 5 \\ 1 \end{pmatrix} + \lambda\begin{pmatrix} 3 \\ 1 \\ -2 \end{pmatrix}\).

1. (a) Show that the equations \(x = -5 + 9\mu, y = 4 + 3\mu, z = 3 - 6\mu\) describe the same line as \(l\).
2. (b) Show that \(\mathbf{r} = (5\mathbf{i} + 7\mathbf{j} - 3\mathbf{k}) + t(-3\mathbf{i} - \mathbf{j} + 2\mathbf{k})\) describes a different line from \(l\).

Two straight tunnels are being planned in a mine. One tunnel passes through the points \(A\) and \(B\), the other through \(C\) and \(D\). Coordinates are in metres. \(A(-1, 9, 0)\), \(B(0, 11, -1)\), \(C(-7, 7, -4)\), \(D(-5, 6, -1)\). Find the coordinates of the point where the two lines cross.

A drone flies along a straight path through waypoint \(P\) and waypoint \(Q\). Coordinates are in metres. A landing marker \(R\) is on the line through \(P\) and \(Q\), with \(PR = 2PQ\). The waypoints are at \(P(-3, 6, 2)\) and \(Q(-1, 3, -2)\).

1. (a) Find a vector equation of the line \(PQ\).
2. (b) Find the possible coordinates of \(R\).

A line has equation \(\mathbf{r} = \begin{pmatrix} 10 \\ -4 \\ 10 \end{pmatrix} + \lambda\begin{pmatrix} 3 \\ -2 \\ 5 \end{pmatrix}\).

1. (a) Find the coordinates of the point where the line meets the \(x\)-axis.
2. (b) Show that the line does not meet the \(y\)-axis.

The lines \(\mathbf{r} = (5\mathbf{j} + 5\mathbf{k}) + \lambda(2\mathbf{i} + \mathbf{j} + 2\mathbf{k})\) and \(\mathbf{r} = (-5\mathbf{i} + 7\mathbf{j} + 9\mathbf{k}) + \mu(\mathbf{i} - \mathbf{j} + p\mathbf{k})\) intersect. Find the value of \(p\) and the coordinates of the point of intersection.

A fishing line \(l\) is modelled as a straight line through the point \(A\), parallel to the vector \(3\mathbf{i} - 4\mathbf{j} + 12\mathbf{k}\). Distances are in centimetres. The point \(A\) has coordinates \((-4, 3, 1)\).

1. (a) Calculate the magnitude of \(3\mathbf{i} - 4\mathbf{j} + 12\mathbf{k}\).
2. (b) Find the possible coordinates of the point \(P\) on \(l\) such that \(AP = 39\).