Vector and Cartesian equations of lines in QCE Specialist Mathematics Unit 3

What this dot point is asking

QCAA wants you to describe a line in three dimensions in three equivalent ways, move between them, and decide how two lines relate. In three dimensions lines need not be parallel or intersecting; they can be skew, missing each other entirely. The assessable skill is setting up the equations correctly and testing for a consistent intersection.

The answer

Vector equation of a line

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

As $t$ varies, $\mathbf{r}$ sweeps out every point on the line. The direction $\mathbf{d}$ may be scaled by any nonzero constant without changing the line.

Parametric equations

Writing $\mathbf{a} = (a_1, a_2, a_3)$ and $\mathbf{d} = (d_1, d_2, d_3)$ and splitting into components gives

These three equations carry the same information as the single vector equation.

Cartesian equation

Eliminating the parameter $t$ (when each $d_i \neq 0$) gives the symmetric Cartesian form

Each equal ratio equals $t$. If a direction component is zero, that coordinate is constant and is stated separately, for example $z = a_3$.

Relationship between two lines

Given $\mathbf{r}_1 = \mathbf{a} + t\mathbf{d}$ and $\mathbf{r}_2 = \mathbf{b} + s\mathbf{e}$, classify them as follows. If $\mathbf{d}$ and $\mathbf{e}$ are scalar multiples, the lines are parallel (and identical if a point of one lies on the other). Otherwise solve the component equations for $t$ and $s$. If a single pair $(t, s)$ satisfies all three components, the lines intersect at that point. If two components give a pair $(t,s)$ that fails the third, the lines are skew: not parallel and not intersecting.

Finding the intersection point

Equate the parametric forms component by component to get three equations in two unknowns $t$ and $s$. Solve any two, then substitute into the third to test consistency. A consistent solution gives the intersection point by substituting $t$ back into $\mathbf{r}_1$.

Why skew lines matter

In two dimensions, non-parallel lines always meet. In three dimensions this fails, so you must check the third equation rather than assuming intersection. This third-equation check is exactly where QCAA distinguishes intersecting from skew lines.

A line from two points

A common starting point is two given points $A$ and $B$ rather than a point and a direction. Take the direction to be the displacement $\vec{AB} = \mathbf{b} - \mathbf{a}$ and the base point to be either $A$ or $B$; both choices describe the same line. This is why a line has many equivalent vector equations: any point on the line and any nonzero scalar multiple of the direction will do. When checking whether a given point lies on a line, substitute its coordinates into the parametric equations and confirm the same value of $t$ satisfies all three.

The angle between two lines

The angle between two lines is the angle between their direction vectors, found with the dot product:

The absolute value gives the acute angle. Two lines are perpendicular when $\mathbf{d}_1 \cdot \mathbf{d}_2 = 0$, even if the lines are skew and never meet, because the angle is defined by directions alone. This distinction (perpendicular directions versus actually meeting) is a regularly tested subtlety.

Distance from a point to a line

To find how far a point $P$ is from a line $\mathbf{r} = \mathbf{a} + t\mathbf{d}$, take the displacement $\vec{AP}$ from the base point, then use the cross product: the perpendicular distance is $\dfrac{|\vec{AP} \times \mathbf{d}|}{|\mathbf{d}|}$. This works because $|\vec{AP} \times \mathbf{d}| = |\vec{AP}||\mathbf{d}|\sin\theta$ is the area of the parallelogram, and dividing by the base length $|\mathbf{d}|$ leaves the height, which is exactly the perpendicular distance.