Components and unit vectors

In 3D, î, ĵ, k̂ are unit vectors along the x, y and z axes. Any vector can be written in component form: r = xî + yĵ + zk̂. The point (x, y, z) gives the components directly.

Magnitude (length), by Pythagoras in 3D: |r| = √(x² + y² + z²). Example: a = 2î – ĵ + 2k̂ has |a| = √(4 + 1 + 4) = 3.

Working with components is easy: everything is done one component at a time.

• Add / subtract: add or subtract matching components. (î + 2ĵ) + (3î – ĵ) = 4î + ĵ.
• Multiply by a number: multiply every component. 3(î – 2k̂) = 3î – 6k̂.
• Equal vectors: all three components must match.

Unit vector in the direction of a: divide a by its own length. â = a/|a|. For a = 2î – ĵ + 2k̂: â = (2/3)î – (1/3)ĵ + (2/3)k̂.

A vector of length k along a is k â. Length 6 along the same a: 6â = 4î – 2ĵ + 4k̂.

Direction ratios of a are just its components (2, –1, 2). Direction cosines l, m, n are the components divided by the magnitude: here 2/3, –1/3, 2/3. They are the cosines of the angles a makes with the three axes.

For direction cosines, always l² + m² + n² = 1. Check: 4/9 + 1/9 + 4/9 = 1.