AS & A Level · Practice questions · Parallel, intersecting and skew lines

Meet, miss or run alongside.

Six original Cambridge-style questions on pairs of lines in three dimensions: parallel, intersecting or skew, with points of intersection and unknown constants.

Original questions All questions on this page are original work, written in the Cambridge AS & A Level style. They are not from past papers. They test the same concepts and skills the syllabus rewards.

The method

Directions first, then three equations.

Parallel? Is one direction a multiple of the other? If so, test one point to decide parallel or the same line.
Otherwise equate general points, with different parameters for the two lines.
Solve two of the three component equations for the two parameters.
Test the third equation: true means intersecting, false means skew.
Point: substitute each parameter into its OWN line, and check both agree.

Recall

[2 marks]

State, with a reason, whether the lines with direction vectors (a) \(2\mathbf{i} - \mathbf{j} + 3\mathbf{k}\) and \(-4\mathbf{i} + 2\mathbf{j} - 6\mathbf{k}\), (b) \(\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}\) and \(2\mathbf{i} + 4\mathbf{j} + 5\mathbf{k}\) are parallel.

(a) \(-4\mathbf{i} + 2\mathbf{j} - 6\mathbf{k} = -2(2\mathbf{i} - \mathbf{j} + 3\mathbf{k})\): parallel. ✓
(b) Factors \(2, 2, \tfrac{5}{3}\) are not all equal: not parallel. ✓

Calculation

[4 marks]

Show that the lines \(\mathbf{r} = (1, 0, 2) + s(1, 1, -1)\) and \(\mathbf{r} = (1, 3, -1) + t(2, -1, 1)\) intersect, and find the position vector of the point of intersection.

Not parallel. \(x\): \(1 + s = 1 + 2t\), so \(s = 2t\). ✓
\(y\): \(s = 3 - t\), so \(2t = 3 - t\), \(t = 1\), \(s = 2\). ✓
\(z\): \(2 - 2 = 0\) and \(-1 + 1 = 0\) ✓. ✓
Point: \(3\mathbf{i} + 2\mathbf{j}\). ✓

Proof

[4 marks]

Show that the lines \(\mathbf{r} = (1, 0, 2) + s(1, 1, -1)\) and \(\mathbf{r} = (2, 1, 4) + t(3, 0, 1)\) are skew.

\((3, 0, 1)\) is not a multiple of \((1, 1, -1)\) (zero \(\mathbf{j}\)-component), so the lines are not parallel. ✓
\(y\): \(s = 1\). \(x\): \(1 + 1 = 2 + 3t\), so \(t = 0\). ✓
\(z\): \(2 - 1 = 1\) but \(4 + 0 = 4\); \(1 \ne 4\). ✓
No common point and not parallel, so the lines are skew. ✓

Calculation

[4 marks]

The lines \(\mathbf{r} = (3, -1, 2) + s(2, 1, -1)\) and \(\mathbf{r} = (5, 1, c) + t(-1, 1, 3)\) intersect. Find \(c\) and the point of intersection.

\(x\): \(3 + 2s = 5 - t\); \(y\): \(-1 + s = 1 + t\), so \(t = s - 2\). ✓
Then \(3 + 2s = 7 - s\), so \(s = \tfrac{4}{3}\) and \(t = -\tfrac{2}{3}\). ✓
\(z\): \(2 - \tfrac{4}{3} = c + 3\left(-\tfrac{2}{3}\right)\), so \(\tfrac{2}{3} = c - 2\) and \(c = \tfrac{8}{3}\). ✓
Point (\(s = \tfrac{4}{3}\) on the first line): \(\left(\tfrac{17}{3}, \tfrac{1}{3}, \tfrac{2}{3}\right)\). ✓

Analysis

[5 marks]

The line \(l\) has equation \(\mathbf{r} = (2, 1, 0) + s(1, -1, 2)\). Determine whether \(l\) is parallel to, intersects, or is skew to each of the lines

\(m\): \(\mathbf{r} = (1, 2, 1) + t(-2, 2, -4)\) and \(n\): \(\mathbf{r} = (6, -3, 8) + u(0, 1, -1)\).

\(m\): \((-2, 2, -4) = -2(1, -1, 2)\), parallel. Is \((1, 2, 1)\) on \(l\)? \(2 + s = 1\) gives \(s = -1\), then \(y = 2\) ✓ and \(z = -2 \ne 1\). So \(l\) and \(m\) are parallel and distinct. ✓
\(n\): not parallel to \(l\). \(x\): \(2 + s = 6\), so \(s = 4\). ✓
\(y\): \(1 - 4 = -3 + u\), so \(u = 0\). \(z\): \(2(4) = 8\) and \(8 - 0 = 8\) ✓. ✓
\(l\) and \(n\) intersect at \((6, -3, 8)\). ✓

Analysis

[6 marks]

A tunnel is being drilled from both ends along straight lines. Relative to a fixed origin, with distances in metres, the first drill moves along \(\mathbf{r} = (0, 0, -10) + s(4, 2, -1)\) and the second along \(\mathbf{r} = (60, 40, -30) + t(-2, -2, 1)\), where \(s, t \ge 0\).

(a) Show that the two bores meet, and find the meeting point. (b) Find the total length of tunnel drilled when they meet, to 3 significant figures.

(a) \(x\): \(4s = 60 - 2t\); \(y\): \(2s = 40 - 2t\). Subtract: \(2s = 20\), so \(s = 10\), \(t = 10\). ✓
\(z\): \(-10 - 10 = -20\) and \(-30 + 10 = -20\) ✓. They meet at \((40, 20, -20)\). ✓
(b) First bore: \(10 \times |(4, 2, -1)| = 10\sqrt{21}\). Second: \(10 \times |(-2, -2, 1)| = 10 \times 3 = 30\). ✓
Total \(= 10\sqrt{21} + 30 = 75.8\) m (3 s.f.). ✓

Mark this once you have attempted all six and checked your working. It records a Practiced badge on the topic and adds a one-time bonus. Revealing the solutions alone does not count.