JEE Main Maths Three Dimensional Geometry 2027: Direction Cosines, Lines & Shortest Distance

Three Dimensional Geometry is 2D coordinate geometry with one extra axis, and most of its ideas are direct extensions of what you already know. The one genuinely new idea, direction cosines, unlocks everything else in the chapter.

Direction cosines are nothing but the coordinates of a unit vector pointing the same way as your line.

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Chapter at a Glance

Priority reflects past-paper trends, not an official NTA weightage.

What the NTA Syllabus Covers

• Coordinates of a point in space, distance between two points, section formula
• Direction ratios and direction cosines, angle between two intersecting lines
• Equation of a line, skew lines, the shortest distance between them and its equation

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Master These Topics

1. Distance, Section Formula and Direction Cosines

Distance between $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$: $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$.

Direction cosines $l, m, n$ of a line are the cosines of the angles it makes with the $x$, $y$ and $z$ axes, and they always satisfy

$$l^2 + m^2 + n^2 = 1$$

If a line has direction ratios $a, b, c$ (any convenient multiple of the direction cosines), then $l = \dfrac{a}{\sqrt{a^2+b^2+c^2}}$, and similarly for $m, n$.

Worked example: For the line from the origin to $P(2, 3, 6)$, the distance is $\sqrt{4+9+36} = 7$, so the direction cosines are $l = \tfrac{2}{7}$, $m = \tfrac{3}{7}$, $n = \tfrac{6}{7}$. Checking: $l^2+m^2+n^2 = \tfrac{4+9+36}{49} = 1$.

Trap: Direction ratios are not unique (any non-zero multiple works), but direction cosines are unique up to an overall sign, since they must satisfy $l^2+m^2+n^2=1$.

2. Angle Between Two Lines

For lines with direction ratios $(a_1,b_1,c_1)$ and $(a_2,b_2,c_2)$:

$$\cos\theta = \frac{a_1a_2+b_1b_2+c_1c_2}{\sqrt{a_1^2+b_1^2+c_1^2}\,\sqrt{a_2^2+b_2^2+c_2^2}}$$

Worked example: Lines with direction ratios $(1,1,2)$ and $(2,-1,1)$: $\cos\theta = \dfrac{2-1+2}{\sqrt{6}\sqrt{6}} = \dfrac{3}{6} = \dfrac{1}{2}$, so $\theta = 60°$.

Lines are perpendicular when $a_1a_2+b_1b_2+c_1c_2=0$, and parallel when their direction ratios are proportional.

3. Equation of a Line

The symmetric form of a line through $(x_1,y_1,z_1)$ with direction ratios $(a,b,c)$ is

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

4. Skew Lines and Shortest Distance

Skew lines are non-parallel lines that never meet. For two lines with position vectors $\vec{a}_1, \vec{a}_2$ on them and direction vectors $\vec{b}_1, \vec{b}_2$, the shortest distance is

$$d = \frac{\left\lvert(\vec{a}_2-\vec{a}_1)\cdot(\vec{b}_1 \times \vec{b}_2)\right\rvert}{\left\lvert \vec{b}_1 \times \vec{b}_2\right\rvert}$$

If $\vec{b}_1 \times \vec{b}_2 = \vec{0}$, the lines are parallel and a different (simpler) distance formula applies instead.

Worked example: Line 1 passes through $(0,0,0)$ with direction $(1,0,0)$, and line 2 passes through $(0,1,1)$ with direction $(0,1,0)$. Here $\vec{b}_1 \times \vec{b}_2 = (1,0,0)\times(0,1,0) = (0,0,1)$, and $\vec{a}_2 - \vec{a}_1 = (0,1,1)$, so $d = \dfrac{\lvert 0+0+1\rvert}{1} = 1$.

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Common Traps to Avoid

• Treating direction ratios as if they were unique like direction cosines.
• Forgetting to check $\vec{b}_1 \times \vec{b}_2 \ne \vec{0}$ before using the skew-line distance formula.
• Mixing up the perpendicularity condition ($a_1a_2+b_1b_2+c_1c_2=0$) with the parallel condition (proportional ratios).
• Using the 2D distance formula and forgetting the $z$-coordinate term.

60-Second Revision Sheet

• $l^2+m^2+n^2=1$; direction cosines from ratios: $l = a/\sqrt{a^2+b^2+c^2}$
• $\cos\theta = \dfrac{a_1a_2+b_1b_2+c_1c_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}$
• Line: $\dfrac{x-x_1}{a}=\dfrac{y-y_1}{b}=\dfrac{z-z_1}{c}$
• Shortest distance between skew lines: $\dfrac{\lvert(\vec{a}_2-\vec{a}_1)\cdot(\vec{b}_1\times\vec{b}_2)\rvert}{\lvert\vec{b}_1\times\vec{b}_2\rvert}$

Your Study Plan

1. Day 1: distance, section formula and direction cosines.
2. Day 2: angle between lines, parallel and perpendicular conditions.
3. Day 3: equation of a line in symmetric and vector form.
4. Day 4: skew lines, shortest distance and a timed mixed set.

Practice Three Dimensional Geometry Questions Free → (opens in a new tab)

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Continue Your Maths Journey

• Previous chapter: Co-ordinate Geometry
• Next chapter: Vector Algebra
• All 14 JEE Main Maths chapters
• Complete JEE Main Syllabus 2027 guide

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