Equation of a Line (Vector & Cartesian)

MATLines in Space

A line in 3D is fixed by one point on it and its direction. The vector form $\vec{r} = \vec{a} + \lambda\vec{b}$ uses a position vector $\vec{a}$ and a direction vector $\vec{b}$ , while the Cartesian (symmetric) form writes the same line as a chain of equal ratios.

ISC requires fluent conversion between the two forms and writing the line through two given points.

Vector equation of a line

\vec{r} = \vec{a} + \lambda\,\vec{b}

\vec{a}

is the position vector of a known point on the line,

\vec{b}

is a direction vector, and

\lambda \in \mathbb{R}

is a scalar parameter.

Cartesian (symmetric) form through a point

\dfrac{x - x_1}{a} = \dfrac{y - y_1}{b} = \dfrac{z - z_1}{c}

(x_1,y_1,z_1)

is a point on the line and

(a,b,c)

are its direction ratios; this common ratio equals the parameter

\lambda

Line through two points (vector form)

\vec{r} = \vec{a} + \lambda(\vec{b} - \vec{a})

\vec{a}, \vec{b}

are the position vectors of the two given points

A

and

B

; the direction is

\vec{b} - \vec{a}

Line through two points (Cartesian form)

\dfrac{x - x_1}{x_2 - x_1} = \dfrac{y - y_1}{y_2 - y_1} = \dfrac{z - z_1}{z_2 - z_1}

A(x_1,y_1,z_1)

and

B(x_2,y_2,z_2)

are the two given points on the line.

To go from Cartesian to vector form: read the point $(x_1,y_1,z_1)$ as $\vec{a}$ and the denominators $(a,b,c)$ as $\vec{b}$ , then write $\vec{r} = \vec{a} + \lambda\vec{b}$ .
To go from vector to Cartesian: write $\vec{r} = x\hat{i}+y\hat{j}+z\hat{k}$ , equate components, and eliminate $\lambda$ to get the equal ratios.
Parallel lines share the same direction ratios (or proportional ones); to write a line parallel to a given line, copy its denominators.
If a direction ratio is $0$ (say $b = 0$ ), write the symmetric form as $\dfrac{x-x_1}{a} = \dfrac{z-z_1}{c},\ y = y_1$ — never put $0$ in a denominator.
The parametric form $x = x_1 + a\lambda,\ y = y_1 + b\lambda,\ z = z_1 + c\lambda$ is useful for finding intersection points and feet of perpendiculars.
Both forms describe the same line for any choice of point on it and any non-zero scalar multiple of the direction vector, so answers may legitimately differ in appearance.
Direction ratios of a line given as the intersection of two planes equal the cross product of the two plane normals.

Where the marks go

Writing a coordinate ratio with $0$ in the denominator instead of stating that coordinate is constant.
Using the wrong point as $\vec{a}$ when two points are given — either point works, but mixing components of both gives a wrong direction.
Forgetting that $\lambda$ is a free real parameter, then treating two different $\lambda$ values as the same point when checking intersection.
Dropping the unit vectors $\hat{i}, \hat{j}, \hat{k}$ in the final vector equation, or leaving the answer as plain coordinate triples.

How the board asks it

Numericalline through two given points
Find the vector and Cartesian equations of the line passing through the points $A(1,-2,3)$ and $B(4,5,-1)$ .
Numericalconversion between vector and cartesian forms
Convert the Cartesian equation of the line $\dfrac{x-2}{3} = \dfrac{y+1}{-2} = \dfrac{z-4}{5}$ into its vector form $\vec{r} = \vec{a} + \lambda\vec{b}$ .
Numericalparallel lines share direction ratios
Find the Cartesian equation of the line that passes through the point $(2,-1,3)$ and is parallel to the line $\vec{r} = (\hat{i}+2\hat{j}-\hat{k}) + \lambda(2\hat{i}-3\hat{j}+4\hat{k})$ .
Numericalzero direction ratio in symmetric form
Write the Cartesian equation of the line through $(3,4,-2)$ having direction ratios $2,0,-5$ , taking care not to place $0$ in a denominator.
Numericaldirection ratios from cross product of plane normals
A line is given as the intersection of the planes $x+2y-z=3$ and $2x-y+z=1$ . Find its direction ratios and hence write its vector equation.
Give reasonssame line under rescaled direction and different point
Assertion–Reason: Assertion (A): The lines $\dfrac{x-1}{2}=\dfrac{y-2}{3}=\dfrac{z-3}{4}$ and $\dfrac{x-3}{4}=\dfrac{y-5}{6}=\dfrac{z-7}{8}$ represent the same line. Reason (R): A line is unchanged when its direction vector is scaled by a non-zero factor and a different point on it is chosen. State whether (A) and (R) are true and whether (R) correctly explains (A).

Practise this topic

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Written for Sublevo. Question text quoted anywhere in these notes is the Council’s and carries its year and paper; the board’s own diagrams are not reproduced.