Intersection of lines and distancesAQA A-Level Further Maths: Flashcards

What these 12 flashcards ask

How do you find the intersection of two lines?
Why must each line use a different parameter?
What are the three possibilities for two lines in 3D?
What are skew lines?
What happens if the third equation fails?
How do you check whether two lines are parallel?
Method for distance from a point to a line?
What is the foot of the perpendicular?
Distance between two parallel lines?
Method for the distance between skew lines?
How do you find the reflection of P in a line?
Can lines in 2D be skew?

Exam questions on Intersection of lines and distances

The line $l_1$ has equation $\mathbf r=\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}+\lambda\begin{pmatrix} 1 \\ -1 \\ 2 \end{pmatrix}$ and the line $l_2$ has equation $\mathbf r=\begin{pmatrix} 1 \\ -1 \\ 4 \end{pmatrix}+\mu\begin{pmatrix} 2 \\ 1 \\ 3 \end{pmatrix}$ .
Show that $l_1$ and $l_2$ intersect and find the coordinates of the point of intersection.2 marks
The line $l_3$ has equation $\mathbf r=\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}+\lambda\begin{pmatrix} 1 \\ -1 \\ 2 \end{pmatrix}$ and the line $l_4$ has equation $\mathbf r=\begin{pmatrix} 1 \\ -1 \\ 5 \end{pmatrix}+\mu\begin{pmatrix} 2 \\ 1 \\ 3 \end{pmatrix}$ .
Explain why $l_3$ and $l_4$ are skew lines.2 marks
The line $l$ has equation $\mathbf r=\begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix}+\lambda\begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix}$ . The point $P$ has coordinates $(4,3,2)$ .
Find the coordinates of the point $F$ on $l$ for which $PF$ is perpendicular to $l$ .3 marks

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).