State the Cartesian equations of a line.

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

Direction of the line through A and B?

\overrightarrow{AB}=\mathbf{b}-\mathbf{a}.

State a line equation using the vector product.

(\mathbf{r}-\mathbf{a})\times\mathbf{b}=\mathbf{0}.

Why does (\mathbf{r}-\mathbf{a})\times\mathbf{b}=\mathbf{0} describe a line?

\mathbf{r}-\mathbf{a} must be parallel to \mathbf{b}, and parallel vectors have zero vector product.

What are direction ratios?

Any numbers l:m:n proportional to the components of a direction vector.

What are direction cosines?

The cosines of the angles between the line and the x-, y- and z-axes: the components of the unit direction vector.

How are direction cosines found from direction ratios?

Divide each ratio by \sqrt{l^2+m^2+n^2}.

State the vector equation of a plane with normal \mathbf{n} through \mathbf{a}.

\mathbf{r}\cdot\mathbf{n}=\mathbf{a}\cdot\mathbf{n}.

How do you find where a line meets a plane?

Substitute the general point of the line into the plane equation, solve for \lambda, and substitute back.

State the formula for the angle between a line and a plane.

\sin\theta=\frac{|\mathbf{d}\cdot\mathbf{n}|}{|\mathbf{d}||\mathbf{n}|}.

State the distance from (x1,y1,z1) to n1x+n2y+n3z=d.

\frac{|n1x1+n2y1+n3z1-d|}{\sqrt{n1^2+n2^2+n3^2}}.

State the distance from a point P to a line through A with direction \mathbf{d}.

\frac{|\overrightarrow{AP}\times\mathbf{d}|}{|\mathbf{d}|}.

Applications of vectors to 3-D geometryEdexcel A-Level Further Maths: Flashcards

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State the vector equation of a line.

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State the vector equation of a line.: $\mathbf{r}=\mathbf{a}+\lambda\mathbf{b}$ , with $\mathbf{a}$ a point on the line and $\mathbf{b}$ its direction.
State the Cartesian equations of a line.: $\frac{x-a_1}{b_1}=\frac{y-a_2}{b_2}=\frac{z-a_3}{b_3}$ .
Direction of the line through $A$ and $B$ ?: $\overrightarrow{AB}=\mathbf{b}-\mathbf{a}$ .
State a line equation using the vector product.: $(\mathbf{r}-\mathbf{a})\times\mathbf{b}=\mathbf{0}$ .
Why does $(\mathbf{r}-\mathbf{a})\times\mathbf{b}=\mathbf{0}$ describe a line?: $\mathbf{r}-\mathbf{a}$ must be parallel to $\mathbf{b}$ , and parallel vectors have zero vector product.
What are direction ratios?: Any numbers $l:m:n$ proportional to the components of a direction vector.
What are direction cosines?: The cosines of the angles between the line and the $x$ -, $y$ - and $z$ -axes: the components of the unit direction vector.
How are direction cosines found from direction ratios?: Divide each ratio by $\sqrt{l^2+m^2+n^2}$ .
What is $\cos^2\alpha+\cos^2\beta+\cos^2\gamma$ ?: $1$ .
State the vector equation of a plane with normal $\mathbf{n}$ through $\mathbf{a}$ .: $\mathbf{r}\cdot\mathbf{n}=\mathbf{a}\cdot\mathbf{n}$ .
How do you find where a line meets a plane?: Substitute the general point of the line into the plane equation, solve for $\lambda$ , and substitute back.
State the formula for the angle between a line and a plane.: $\sin\theta=\frac{|\mathbf{d}\cdot\mathbf{n}|}{|\mathbf{d}||\mathbf{n}|}$ .
State the distance from $(x_1,y_1,z_1)$ to $n_1x+n_2y+n_3z=d$ .: $\frac{|n_1x_1+n_2y_1+n_3z_1-d|}{\sqrt{n_1^2+n_2^2+n_3^2}}$ .
State the distance from a point $P$ to a line through $A$ with direction $\mathbf{d}$ .: $\frac{|\overrightarrow{AP}\times\mathbf{d}|}{|\mathbf{d}|}$ .

Exam questions on Applications of vectors to 3-D geometry

The line $l$ has vector equation $\mathbf{r}=(\mathbf{i}-2\mathbf{j}+3\mathbf{k})+\lambda(2\mathbf{i}+\mathbf{j}-2\mathbf{k})$ .
Find Cartesian equations for $l$ .2 marks
The plane $\Pi$ passes through the point $A(1,2,3)$ and is perpendicular to the vector $2\mathbf{i}-\mathbf{j}+2\mathbf{k}$ .
The line $m$ passes through the origin and has direction $\mathbf{i}+\mathbf{j}+\mathbf{k}$ . Find the coordinates of the point where $m$ meets $\Pi$ .2 marks
The line $l$ has Cartesian equation $\frac{x-1}{3}=\frac{y+2}{-4}=\frac{z}{12}$ , and the point $P$ has coordinates $(5,1,0)$ .
Find the direction cosines of $l$ , and the acute angle that $l$ makes with the $z$ -axis.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).