Vectors: basic concepts

The essentials

A vector is a directed line segment: it has a length (its magnitude) and a direction. A quantity with magnitude only, such as mass or temperature, is a scalar. If a point P has coordinates (x, y, z), its position vector from the origin is OP = xi + yj + zk, with magnitude |OP| = √(x² + y² + z²).

If a vector makes angles α, β, γ with the positive x-, y- and z-axes, then l = cos α, m = cos β, n = cos γ are its direction cosines. For the vector xi + yj + zk of magnitude r, l = x/r, m = y/r, n = z/r, and always l² + m² + n² = 1. Any three numbers a, b, c proportional to l, m, n are called direction ratios.

Key points

• Zero vector: magnitude 0, direction not defined.
• Unit vector: magnitude 1. The unit vector along a is a/|a|.
• Equal vectors: same magnitude and same direction, wherever they start.
• Collinear (parallel) vectors lie along the same or parallel lines.
• Coinitial vectors share the same starting point.
• Negative of a vector: same magnitude, opposite direction.
• If A and B have position vectors a and b, then AB = b − a.

Exam tips

• To get direction cosines from direction ratios a, b, c, divide each by √(a² + b² + c²).
• Two angles of a vector with the axes determine the third (up to a choice of sign) through cos²α + cos²β + cos²γ = 1.

Common mistakes

• Forgetting to divide by the magnitude when asked for direction cosines.
• Writing AB = a − b; the vector from A to B is (position of B) − (position of A).