Finding the Equation of a Line

Vector and Cartesian form

A line through the point with position vector $\vec{a}$ , parallel to the direction vector $\vec{b}$ , has vector equation $\vec{r} = \vec{a} + \lambda\vec{b}$ . With $\vec{a} = (x_1,y_1,z_1)$ and $\vec{b} = (a,b,c)$ , the Cartesian form is:

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

Worked example: through two points

Find the line through $(3,-2,-5)$ and $(3,-2,6)$ . The direction ratios are $(3-3, -2-(-2), 6-(-5)) = (0,0,11)$ :

\vec{r} = (3\hat{i}-2\hat{j}-5\hat{k}) + \lambda(11\hat{k}), \qquad \dfrac{x-3}{0} = \dfrac{y+2}{0} = \dfrac{z+5}{11}

A zero denominator means that coordinate stays fixed: $x = 3$ and $y = -2$ while $z$ varies, a line parallel to the $z$ -axis.

Worked example: point and parallel vector

The line through $2\hat{i}+3\hat{j}+\hat{k}$ parallel to $4\hat{i}-2\hat{j}+3\hat{k}$ is:

\vec{r} = (2+4\lambda)\hat{i}+(3-2\lambda)\hat{j}+(1+3\lambda)\hat{k}

At $\lambda = 1$ it passes through $(6, 1, 4)$ , a quick way to generate a second point for checking.

Worked example: is a point on the line?

Does $(3,-2,10)$ lie on the first line? Its $x$ and $y$ match the fixed values 3 and −2, and $z + 5 = 15 = 11\lambda$ gives $\lambda = \dfrac{15}{11}$ . One value of $\lambda$ works for every coordinate, so yes.

Worked example: angle between two lines

Lines with direction ratios $(1,2,2)$ and $(2,-1,2)$ meet at an angle $\theta$ with:

\cos\theta = \dfrac{|1(2)+2(-1)+2(2)|}{\sqrt{1+4+4}\,\sqrt{4+1+4}} = \dfrac{4}{9}, \qquad \theta \approx 63.6^\circ

Common mistakes

Sign slips with negative coordinates. Through $(3,-2,-5)$ the Cartesian form has $y+2$ and $z+5$ , not $y-2$ and $z-5$ .
Reading a zero denominator as division by zero. It's shorthand for that coordinate being constant.
Checking a point with different λ values. A point is on the line only if one λ satisfies every coordinate.
Confusing direction ratios and direction cosines. Direction cosines are the ratios divided by the vector's length.

Where this comes up in exams

The 2025-26 CBSE Class XII syllabus lists the Cartesian and vector equation of a line, skew lines, the shortest distance between two lines and the angle between two lines. JEE Main 2026 lists the equation of a line and skew lines in three-dimensional geometry, and JEE Advanced 2026 lists the equation of a straight line in space.

Checked against CBSE Mathematics curriculum, Classes XI-XII (2025-26), JEE (Main) 2026 syllabus (NTA) and JEE (Advanced) 2026 syllabus. Syllabi can change from year to year, so confirm with the latest official notification for your exam.

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