Intersections and distances in three dimensionsEdexcel A-Level Further Maths: Flashcards

What these 13 flashcards ask

Condition for a line \mathbf{r}=\mathbf{a}+t\mathbf{d} to be parallel to a plane with normal \mathbf{n}?
How do you find where a line meets a plane?
A line is parallel to a plane and \mathbf{a}\cdot\mathbf{n}=k. What does this mean?
Distance from (\alpha,\beta,\gamma) to n1x+n2y+n3z+d=0?
Distance from (3,-1,4) to 2x-2y+z=7?
How do you find the foot of the perpendicular from a point to a plane?
Reflection of A in a plane, given the foot F?
Distance from a point P to the line \mathbf{r}=\mathbf{a}+\lambda\mathbf{d} by cross product?
Condition used to find the foot of the perpendicular from P to a line?
What are skew lines?
Shortest distance between skew lines \mathbf{r}=\mathbf{a}1+s\mathbf{d}1 and \mathbf{r}=\mathbf{a}2+t\mathbf{d}2?
How do you show two lines are skew?
Distance between two parallel lines?

Exam questions on Intersections and distances in three dimensions

The line $l$ has equation $\mathbf{r}=\begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix}+t\begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix}$ and the plane $\Pi$ has equation $2x-y+3z=14$ .
The point $A$ with position vector $(1,0,2)$ lies on $l$ . Find the exact distance from $A$ to the point where $l$ meets $\Pi$ .2 marks
The plane $\Pi$ has equation $2x-2y+z=7$ and the point $A$ has coordinates $(3,-1,4)$ .
Find the position vector of the reflection of $A$ in $\Pi$ .2 marks
The line $l$ has equation $\mathbf{r}=\begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix}+\lambda\begin{pmatrix} 2 \\ 1 \\ -2 \end{pmatrix}$ and the point $P$ has coordinates $(4,2,0)$ .
Find the coordinates of the foot of the perpendicular from $P$ 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).