Three Dimensional Geometry – JEE Mains Mathematics

• by Zenith LMS
• Updated April 2025

1. Coordinates of a Point in Space

• In three-dimensional space, a point is represented by an ordered triplet (x, y, z), where:
• x, y, and z represent the coordinates of the point along the x, y, and z axes respectively.

2. Distance Between Two Points

• The distance between two points P₁(x₁, y₁, z₁) and P₂(x₂, y₂, z₂) in three-dimensional space is given by:
• Distance = √[(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²]

3. Section Formula

• The section formula in three dimensions is used to find the coordinates of a point dividing the line segment joining two points in a given ratio.
• If the point P divides the line joining points A(x₁, y₁, z₁) and B(x₂, y₂, z₂) in the ratio m:n, the coordinates of P are:
• P = ((mx₂ + nx₁) / (m + n), (my₂ + ny₁) / (m + n), (mz₂ + nz₁) / (m + n))

4. Direction Ratios and Direction Cosines

• The **direction ratios** (dr) of a line are the proportional values of the differences in the coordinates of two points on the line, i.e., (x₂ - x₁), (y₂ - y₁), (z₂ - z₁).
• The **direction cosines** (l, m, n) of a line are the cosines of the angles made by the line with the positive x, y, and z-axes respectively.
• l = cos(α), m = cos(β), n = cos(γ), where α, β, and γ are the angles the line makes with the x, y, and z-axes respectively.

5. Angle Between Two Intersecting Lines

• The angle θ between two lines with direction ratios (l₁, m₁, n₁) and (l₂, m₂, n₂) is given by:
• cos(θ) = (l₁l₂ + m₁m₂ + n₁n₂) / (√(l₁² + m₁² + n₁²) * √(l₂² + m₂² + n₂²))

6. Equation of a Line

• The equation of a line in three-dimensional space passing through a point (x₁, y₁, z₁) with direction ratios (l, m, n) is given by:
• (x - x₁) / l = (y - y₁) / m = (z - z₁) / n

7. Skew Lines

• Two lines are said to be **skew lines** if they are not parallel and do not intersect.
• The shortest distance between two skew lines can be found using the formula:
• Distance = |(r₂ - r₁) . (d₁ × d₂)| / |d₁ × d₂|
• Where r₁ and r₂ are position vectors of points on the respective lines, and d₁ and d₂ are direction ratios of the lines.

8. Shortest Distance Between Two Skew Lines

• The shortest distance between two skew lines can be calculated using the cross product of their direction ratios, as explained in the previous section on skew lines.