Vector and Cartesian equations of lines and planes, the scalar and vector products, angles between lines and planes, intersections and shortest distances in three dimensions.

What this dot point is asking

AQA wants you to write lines and planes in vector and Cartesian form, use the scalar and vector products, find the angle between two lines, a line and a plane, or two planes, locate intersections, and compute the shortest distance from a point to a line or plane and between skew lines.

Lines and planes

A line through $\mathbf{a}$ with direction $\mathbf{d}$ is $\mathbf{r} = \mathbf{a} + \lambda \mathbf{d}$. A plane can be written as $\mathbf{r} = \mathbf{a} + \lambda \mathbf{u} + \mu \mathbf{v}$, or more usefully in scalar product form $\mathbf{r} \cdot \mathbf{n} = d$, where $\mathbf{n}$ is normal to the plane and $d = \mathbf{a} \cdot \mathbf{n}$.

The scalar and vector products

Angles

The angle between two lines comes from the scalar product of their directions. The angle $\phi$ between a line of direction $\mathbf{d}$ and a plane of normal $\mathbf{n}$ satisfies $\sin\phi = \frac{|\mathbf{d} \cdot \mathbf{n}|}{|\mathbf{d}||\mathbf{n}|}$, because the line makes angle $90^\circ - \phi$ with the normal.

Intersections

To find where two lines meet, set their vector equations equal and solve the resulting simultaneous equations component by component. Two of the three equations fix the parameters $\lambda$ and $\mu$; the third is a consistency check. If it holds, the lines intersect at the point found by substituting back; if it fails, the lines are skew (assuming they are not parallel). A line meets a plane $\mathbf{r}\cdot\mathbf{n} = d$ where its parameter value satisfies $(\mathbf{a} + \lambda\mathbf{d})\cdot\mathbf{n} = d$; solving the single linear equation for $\lambda$ and substituting gives the point of intersection. Two non-parallel planes intersect in a line whose direction is $\mathbf{n}_1\times\mathbf{n}_2$.

Distances

The shortest distance from a point $P$ to a plane $\mathbf{r} \cdot \mathbf{n} = d$ is $\frac{|\mathbf{p} \cdot \mathbf{n} - d|}{|\mathbf{n}|}$. The shortest distance between two skew lines uses the common perpendicular direction $\mathbf{d}_1 \times \mathbf{d}_2$, projected onto the vector joining a point on each line.

Across a typical question you will move between forms: write a line from two points, convert a plane between parametric and scalar-product form using the normal $\mathbf{n} = \mathbf{u}\times\mathbf{v}$, then compute an angle or distance. Keep the line-plane angle on $\sin$ with the normal, always divide by the relevant magnitudes, and test for skew lines by checking the third simultaneous equation rather than assuming an intersection exists.