Three Dimensional Geometry Mind Map

Direction cosines of a line are the cosines of the angles made by the line with the positive x-, y-, and z-axes, usually denoted by l, m, and n. Direction ratios are any three numbers proportional to the direction cosines.

When the question gives direction ratios, first divide by √(a² + b² + c²) if direction cosines are required.

A line in space is determined by one point on the line and one direction vector. Its vector form is r = a + λb, where a is the position vector of a fixed point and b is a direction vector.

Before writing the equation, underline the point and circle the direction vector or direction ratios. This prevents mixing their roles.

The angle between two lines in space is the angle between their direction vectors. If the direction vectors are a₁ and a₂, the acute angle θ between the lines is found using their dot product.

For angle questions, ignore the fixed point of each line after confirming the direction vectors. The direction vectors alone decide the angle.

The shortest distance between two skew lines is the length of the common perpendicular between them. Skew lines are non-parallel, non-intersecting lines in three-dimensional space.

Before applying the skew-lines formula, check that b₁ × b₂ is not the zero vector. If it is zero, the lines are parallel and require the parallel-lines distance method.