Direction Cosines and Direction Ratios

Identify whether the given numbers are ratios or cosines. For ratios, calculate √(a²+b²+c²). Divide each ratio by this value. For cosines, check the identity l² + m² + n² = 1. If one cosine is missing, substitute the known values and solve carefully with the correct sign condition if given.

A simple 3D axes sketch may help show the angles α, β, γ with the coordinate axes, but most exam questions can be solved using the formula directly.

Writing 3, 4, 12 as direction cosines without normalising, which fails because 3² + 4² + 12² is not 1.

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.

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. Key rule: if l, m, n are direction cosines, then l² + m² + n² = 1. If a, b, c are direction ratios, then direction cosines are ±a/√(a²+b²+c²), ±b/√(a²+b²+c²), ±c/√(a²+b²+c²), with sign chosen according to the line's direction. For direction ratios 2, -1, 2, the magnitude factor is √(2² + (-1)² + 2²) = 3. The direction cosines are 2/3, -1/3, 2/3.

If a line makes angles α, β, and γ with the coordinate axes, then l = cos α, m = cos β, and n = cos γ. Since these come from a unit direction vector, they always satisfy l² + m² + n² = 1. Direction ratios a, b, c only show direction proportion, so they need not satisfy this identity until converted into direction cosines. Key rule: if l, m, n are direction cosines, then l² + m² + n² = 1. If a, b, c are direction ratios, then direction cosines are ±a/√(a²+b²+c²), ±b/√(a²+b²+c²), ±c/√(a²+b²+c²), with sign chosen according to the line's direction. Given direction ratios 3, 4, 12, find direction cosines. Compute √(3² + 4² + 12²) = √(9 + 16 + 144) = √169 = 13. Therefore l = 3/13, m = 4/13, n = 12/13. Check: 9/169 + 16/169 + 144/169 = 169/169 = 1. Usually asked as a short answer problem: convert direction ratios to direction cosines, verify whether given numbers can be direction cosines, or find a missing direction cosine using l² + m² + n² = 1. Writing 3, 4, 12 as direction cosines without normalising, which fails because 3² + 4² + 12² is not 1.