What does (\mathbf{r}-\mathbf{a})\times\mathbf{b}=\mathbf{0} represent?

The line through \mathbf{a} parallel to \mathbf{b}.

Cartesian form of a line?

\frac{x-a1}{b1}=\frac{y-a2}{b2}=\frac{z-a3}{b3}

Vector equation of a plane with normal \mathbf{n}?

\mathbf{r}\cdot\mathbf{n}=p with p=\mathbf{a}\cdot\mathbf{n}

Parametric form of a plane?

\mathbf{r}=\mathbf{a}+s\mathbf{b}+t\mathbf{c}

Normal to \mathbf{r}=\mathbf{a}+s\mathbf{b}+t\mathbf{c}?

\mathbf{n}=\mathbf{b}\times\mathbf{c}

Distance from (x1,y1,z1) to \alpha x+\beta y+\gamma z=p?

\frac{|\alpha x1+\beta y1+\gamma z1-p|}{\sqrt{\alpha^2+\beta^2+\gamma^2}}

How do you find a point on the line of intersection?

Set one coordinate (e.g. z=0) and solve the other two equations.

Lines that are not parallel and do not intersect.

Shortest distance between skew lines?

\frac{\left|(\mathbf{a}2-\mathbf{a}1)\cdot(\mathbf{b}1\times\mathbf{b}2)\right|}{\left|\mathbf{b}1\times\mathbf{b}2\right|}

Lines and planes in three dimensionsEdexcel International A Level Further Maths: Flashcards

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Vector equation of a line?

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Vector equation of a line?: $\mathbf{r}=\mathbf{a}+\lambda\mathbf{b}$
What does $(\mathbf{r}-\mathbf{a})\times\mathbf{b}=\mathbf{0}$ represent?: The line through $\mathbf{a}$ parallel to $\mathbf{b}$ .
Cartesian form of a line?: $\frac{x-a_1}{b_1}=\frac{y-a_2}{b_2}=\frac{z-a_3}{b_3}$
Vector equation of a plane with normal $\mathbf{n}$ ?: $\mathbf{r}\cdot\mathbf{n}=p$ with $p=\mathbf{a}\cdot\mathbf{n}$
Parametric form of a plane?: $\mathbf{r}=\mathbf{a}+s\mathbf{b}+t\mathbf{c}$
Normal to $\mathbf{r}=\mathbf{a}+s\mathbf{b}+t\mathbf{c}$ ?: $\mathbf{n}=\mathbf{b}\times\mathbf{c}$
Cartesian form of $\mathbf{r}\cdot(\alpha\mathbf{i}+\beta\mathbf{j}+\gamma\mathbf{k})=p$ ?: $\alpha x+\beta y+\gamma z=p$
Distance from $(x_1,y_1,z_1)$ to $\alpha x+\beta y+\gamma z=p$ ?: $\frac{|\alpha x_1+\beta y_1+\gamma z_1-p|}{\sqrt{\alpha^2+\beta^2+\gamma^2}}$
Direction of the line of intersection of two planes?: $\mathbf{n}_1\times\mathbf{n}_2$
How do you find a point on the line of intersection?: Set one coordinate (e.g. $z=0$ ) and solve the other two equations.
What are skew lines?: Lines that are not parallel and do not intersect.
Shortest distance between skew lines?: $\frac{\left|(\mathbf{a}_2-\mathbf{a}_1)\cdot(\mathbf{b}_1\times\mathbf{b}_2)\right|}{\left|\mathbf{b}_1\times\mathbf{b}_2\right|}$

Exam questions on Lines and planes in three dimensions

The plane $\Pi$ has equation $\mathbf{r}\cdot(2\mathbf{i}-\mathbf{j}+2\mathbf{k})=6$ and the point $A$ has coordinates $(4,1,3)$ .
Write down an equation of $\Pi$ in the form $\mathbf{r}=\mathbf{a}+s\mathbf{b}+t\mathbf{c}$ .2 marks
The line $l$ has equation $\left(\mathbf{r}-(\mathbf{i}+2\mathbf{j}-\mathbf{k})\right)\times(2\mathbf{i}-\mathbf{j}+3\mathbf{k})=\mathbf{0}$ .
Find the coordinates of the point where $l$ meets the plane $y=0$ .2 marks
The planes $\Pi_1$ and $\Pi_2$ have equations $x+2y-z=4$ and $2x-y+3z=3$ respectively.
Find a vector equation of the line of intersection $l$ of $\Pi_1$ and $\Pi_2$ .3 marks

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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).