CLASS 12 MATHS · THREE DIMENSIONAL GEOMETRY

Direction cosines and distances in 3D

Direction ratios a, b, c give direction cosines a/r, b/r, c/r with r = √(a² + b² + c²). Distance from (x₀, y₀, z₀) to ax + by + cz + d = 0 is |ax₀ + by₀ + cz₀ + d|/√(a² + b² + c²).

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Worked examples

Example 1 · level 1

A line has direction ratios −12, 4, −3. Find its direction cosine n (with the z-axis), taking the same signs as the direction ratios.

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r = √(144 + 16 + 9) = 13.
Direction cosines: −12/13, 4/13, −3/13.
Required = −3/13.

Answer: -3/13

Example 2 · level 2

Find the distance between the points A(1, −4, 2) and B(2, 0, −6).

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AB = √(1 + 16 + 64) = √81 = 9.

Answer: 9

Example 3 · level 3

Find the distance of the point (1, −2, 3) from the plane −4x − 4y + 7z − 10 = 0.

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Numerator = |15| = 15.
Denominator = 9.
Distance = 5/3 units.

Answer: 5/3