Vector equations of lines and planes (WACE Year 12 Mathematics Specialist Unit 4)

What this dot point is asking

SCSA Unit 4 uses three-dimensional vectors to describe lines and planes. You must write each object in vector, parametric and cartesian form, decide whether lines intersect, are parallel or are skew, find points of intersection, and compute the angles between lines, between planes, and between a line and a plane.

Why a line needs a point and a direction

A line in three dimensions is fixed by two pieces of information: a single point it passes through and the direction it runs in. The point anchors the line in space, and the direction vector says which way it goes; every other point on the line is reached by travelling some multiple of the direction vector from the anchor point. This is the meaning of $\mathbf{r} = \mathbf{a} + \lambda\mathbf{d}$, where $\lambda$ is the signed number of direction-vector steps from $\mathbf{a}$. A plane similarly needs a point and a normal direction, the normal pinning down the plane's tilt because every vector lying in the plane must be perpendicular to it. Understanding why each object needs exactly this data makes the equations easy to recall and to build from given information.

Vector and parametric equations of a line

A line through point $A$ with position vector $\mathbf{a}$ and direction vector $\mathbf{d}$ is

Splitting into components gives the parametric form $x = a_1 + \lambda d_1$, $y = a_2 + \lambda d_2$, $z = a_3 + \lambda d_3$.

Two lines are parallel if their direction vectors are scalar multiples. If they are not parallel, they either intersect (a common point exists) or are skew (no common point and not parallel).

Equations of a plane

A plane with normal vector $\mathbf{n}$ passing through a point with position vector $\mathbf{a}$ satisfies

If a plane is given by three points, find two direction vectors in the plane and take their cross product to obtain the normal $\mathbf{n}$.

Angles

Use the dot product, $\mathbf{u}\cdot\mathbf{v} = |\mathbf{u}||\mathbf{v}|\cos\phi$, to extract angles. Take the acute angle by using the modulus of the dot product.

• Between two lines: angle between their direction vectors, $\cos\theta = \dfrac{|\mathbf{d_1}\cdot\mathbf{d_2}|}{|\mathbf{d_1}||\mathbf{d_2}|}$.
• Between two planes: angle between their normals.
• Between a line and a plane: if $\alpha$ is the angle between the direction $\mathbf{d}$ and the normal $\mathbf{n}$, the line-plane angle is $90^\circ - \alpha$, so $\sin\theta = \dfrac{|\mathbf{d}\cdot\mathbf{n}|}{|\mathbf{d}||\mathbf{n}|}$.

Intersections

To intersect a line with a plane, substitute the parametric coordinates into the plane equation and solve for $\lambda$, then back-substitute. To intersect two lines, equate components and solve; if the resulting equations are inconsistent the lines are skew (assuming they are not parallel).

Distances

The distance from a point with position vector $\mathbf{p}$ to a plane $\mathbf{r}\cdot\mathbf{n} = k$ is $\dfrac{|\mathbf{p}\cdot\mathbf{n} - k|}{|\mathbf{n}|}$. The shortest distance from a point to a line uses the perpendicular component: project the point-to-line vector onto $\mathbf{d}$ and subtract.