Lines in three dimensions
Vectors
- A2
- Non-calculator
- 8 questions
- 37 marks
These are original practice questions for the Edexcel A-level Pure content, not a past paper. This site is independent and not affiliated with Pearson.
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Write down a vector equation of the line through $(1, 2, 3)$ in the direction of $\begin{pmatrix} 2 \\ -1 \\ 4 \end{pmatrix}$.
3 marks
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Decide whether the point $(5, 0, 11)$ lies on the line $\mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ -1 \\ 4 \end{pmatrix}$.
3 marks
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Show that the lines $\mathbf{r} = \begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}$ and $\mathbf{r} = \begin{pmatrix} 0 \\ 1 \\ 2 \end{pmatrix} + \mu \begin{pmatrix} 2 \\ 2 \\ 0 \end{pmatrix}$ are parallel and distinct.
4 marks
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Find the intersection of the lines $\mathbf{r} = \begin{pmatrix} 2 \\ 1 \\ 0 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}$ and $\mathbf{r} = \begin{pmatrix} 4 \\ 5 \\ 2 \end{pmatrix} + \mu \begin{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix}$.
4 marks
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Determine whether the lines $\mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 1 \\ 1 \end{pmatrix}$ and $\mathbf{r} = \begin{pmatrix} 3 \\ 0 \\ 2 \end{pmatrix} + \mu \begin{pmatrix} 2 \\ 1 \\ 1 \end{pmatrix}$ intersect, are parallel and distinct, or are skew.
5 marks
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Show that the lines $\mathbf{r} = \begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 1 \\ -1 \end{pmatrix}$ and $\mathbf{r} = \begin{pmatrix} 2 \\ 1 \\ 0 \end{pmatrix} + \mu \begin{pmatrix} 1 \\ -1 \\ 1 \end{pmatrix}$ are skew.
5 marks
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Find the point of intersection of the lines $\mathbf{r} = \begin{pmatrix} 0 \\ 1 \\ 2 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}$ and $\mathbf{r} = \begin{pmatrix} 2 \\ 0 \\ 2 \end{pmatrix} + \mu \begin{pmatrix} -1 \\ 2 \\ 1 \end{pmatrix}$.
6 marks
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Show that the lines $\mathbf{r} = \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix}$ and $\mathbf{r} = \begin{pmatrix} 2 \\ 3 \\ -1 \end{pmatrix} + \mu \begin{pmatrix} 1 \\ -1 \\ 1 \end{pmatrix}$ are skew.
7 marks
Mark scheme
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Answer. $\mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ -1 \\ 4 \end{pmatrix}$
- Use a point on the line as the position vector and the given direction as the direction vector.
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Answer. Yes. It is the point where $\lambda = 2$.
- $1 + 2\lambda = 5$ gives $\lambda = 2$.
- The $y$ and $z$ coordinates also match when $\lambda = 2$.
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Answer. The lines are parallel and distinct.
- The second direction vector is twice the first, so the lines are parallel.
- The point $(0, 1, 2)$ does not lie on the first line.
- Parallel lines that are not the same line do not meet.
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Answer. $(4, 5, 2)$
- Equate components: $2 + \lambda = 4 + \mu$, $1 + 2\lambda = 5$ and $\lambda = 2 - \mu$.
- The $y$ equation gives $\lambda = 2$.
- Then $\mu = 0$, and both lines pass through $(4, 5, 2)$.
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Answer. The lines are parallel and distinct.
- The direction vectors are equal, so the lines are parallel.
- Substituting the point $(3, 0, 2)$ into the first line gives inconsistent $y$ coordinates.
- The lines are not the same, so they do not intersect.
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Answer. The lines are skew.
- The direction vectors are not scalar multiples, so the lines are not parallel.
- Equating components gives $\lambda - \mu = 1$ and $\lambda + \mu = 1$, so $\lambda = 1$ and $\mu = 0$.
- These values do not satisfy the $z$ equation $2 - \lambda = \mu$.
- Non-parallel lines that do not meet are skew.
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Answer. $(1, 2, 3)$
- Equate components: $\lambda = 2 - \mu$, $1 + \lambda = 2\mu$ and $2 + \lambda = 2 + \mu$.
- The $z$ equation gives $\lambda = \mu$.
- Then $\lambda = 1$ and $\mu = 1$, and the point is $(1, 2, 3)$.
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Answer. The lines are skew.
- The directions $\begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix}$ and $\begin{pmatrix} 1 \\ -1 \\ 1 \end{pmatrix}$ are not parallel.
- Equating $x$ and $y$ gives $\lambda = 2$ and $\mu = 1$.
- These values give $z = 5$ on the first line and $z = 0$ on the second.
- The lines are not parallel and do not meet, so they are skew.