Three Dimensional Geometry | Class 12 Mathematics Notes

11.2.1 Direction cosines of a line passing through two points

Given two distinct points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) in space, exactly one line passes through them. The direction cosines of this line can be found by considering the differences in coordinates. The length PQ is the distance between points P and Q, calculated as √[(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²]. The direction cosines l, m, n of the line PQ are the cosines of the angles the line makes with the x, y, and z axes respectively. Using right-angled triangle properties, cos α = (x₂ - x₁)/PQ, cos β = (y₂ - y₁)/PQ, and cos γ = (z₂ - z₁)/PQ. Hence, the direction cosines are the normalized differences in coordinates. Direction ratios can be taken as (x₂ - x₁), (y₂ - y₁), (z₂ - z₁) or their negatives. Several examples illustrate these calculations, including finding direction cosines from given angles, direction ratios, and points. The direction cosines of coordinate axes are also given: x-axis (1, 0, 0), y-axis (0, 1, 0), and z-axis (0, 0, 1). Additionally, collinearity of points is checked by comparing proportionality of direction ratios of segments formed by the points.

📊 Diagram: Fig 11.2; Let l, m, n be the direction cosines of the line PQ and let it makes angles α, β and γ with the x, y and z -axis, respectively.

🔗 Connection: Leads to the study of vector and Cartesian equations of lines in space.