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

The points \(A(2, -3, 1)\), \(B(5, -1, -3)\) and \(C(-4, -7, 9)\) 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 \((3, -2, 4)\) with direction \(\begin{pmatrix} 2 \\ 1 \\ -5 \end{pmatrix}\).
2. (b) The line through the point \((-1, 4, 2)\) with direction \(3\mathbf{i} - 2\mathbf{j} + \mathbf{k}\).
3. (c) The line through the point \((2, 6)\) with direction \(\begin{pmatrix} -3 \\ 5 \end{pmatrix}\).

Write each line in the form asked.

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

A line \(l\) has equation \(\mathbf{r} = (2\mathbf{i} + 5\mathbf{j} - \mathbf{k}) + \lambda(3\mathbf{i} - \mathbf{j} + 2\mathbf{k})\). Find parametric equations of the line parallel to \(l\) that passes through the point \((4, -1, 3)\).

A line has parametric equations \(x = 4 - \lambda, y = 2 + 3\lambda, z = -5 + 2\lambda\).

1. (a) Determine whether the point \((6, -4, -9)\) lies on the line.
2. (b) The point \((1, p, q)\) lies on the line. Find \(p\) and \(q\).

Find the point of intersection of the lines \(\mathbf{r} = \begin{pmatrix} 1 \\ -8 \\ 9 \end{pmatrix} + \lambda\begin{pmatrix} 1 \\ 3 \\ -2 \end{pmatrix}\) and \(\mathbf{r} = (5\mathbf{i} - 3\mathbf{j} + 9\mathbf{k}) + \mu(2\mathbf{i} - \mathbf{j} + 4\mathbf{k})\).

Decide about the lines in each part.

1. (a) Show that the lines \(\mathbf{r} = \begin{pmatrix} -1 \\ 3 \\ 1 \end{pmatrix} + \lambda\begin{pmatrix} 2 \\ 1 \\ -3 \end{pmatrix}\) and \(\mathbf{r} = \begin{pmatrix} 3 \\ -2 \\ -3 \end{pmatrix} + \mu\begin{pmatrix} -1 \\ 3 \\ 2 \end{pmatrix}\) do not intersect.
2. (b) The lines \(l_1\) and \(l_2\) have parametric equations \(l_1: x = -1 + 3\lambda, y = -3 + 2\lambda, z = 7 - \lambda\) and \(l_2: x = 4 + \mu, y = -3 - \mu, z = 14 + 4\mu\). Show that they intersect and find the point of intersection.

The line \(l\) has equation \(\mathbf{r} = \begin{pmatrix} 3 \\ 0 \\ -4 \end{pmatrix} + \lambda\begin{pmatrix} 1 \\ -2 \\ 3 \end{pmatrix}\).

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

A laser show uses two straight beams. One beam passes through the points \(A\) and \(B\), the other through \(C\) and \(D\). Coordinates are in metres. \(A(0, 4, -7)\), \(B(2, 3, -4)\), \(C(7, 8, -7)\), \(D(8, 10, -9)\). Find the coordinates of the point where the two lines cross.

A zip wire runs in a straight line from post \(P\) to post \(Q\). Coordinates are in metres. A rest platform \(R\) is on the line through \(P\) and \(Q\), with \(PR = 3PQ\). The posts are at \(P(2, -1, 4)\) and \(Q(5, 1, 0)\).

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

A line has equation \(\mathbf{r} = \begin{pmatrix} -6 \\ -16 \\ 9 \end{pmatrix} + \lambda\begin{pmatrix} 2 \\ 7 \\ -3 \end{pmatrix}\).

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

The lines \(\mathbf{r} = \begin{pmatrix} 3 \\ 5 \\ -4 \end{pmatrix} + \lambda\begin{pmatrix} 1 \\ -2 \\ 3 \end{pmatrix}\) and \(\mathbf{r} = \begin{pmatrix} 7 \\ -5 \\ -6 \end{pmatrix} + \mu\begin{pmatrix} -1 \\ 3 \\ p \end{pmatrix}\) intersect. Find the value of \(p\) and the coordinates of the point of intersection.

A cable \(l\) passes through the point \(A\) and is parallel to the vector \(2\mathbf{i} + 6\mathbf{j} - 9\mathbf{k}\). Distances are in metres. The point \(A\) has coordinates \((3, -1, 2)\).

1. (a) Calculate the magnitude of \(2\mathbf{i} + 6\mathbf{j} - 9\mathbf{k}\).
2. (b) Find the possible coordinates of the point \(P\) on \(l\) such that \(AP = 44\).