Lesson 1 of 11 · 8 min

Scalars, vectors and position vectors

NCERT §10.1–10.2

The science club's survey drone hovers over the playground, and the pilot calls out 'It is 13 metres from the pad.' True, but if you look up, where exactly do you look?

The story this chapter follows: The school survey drone

The science club flies a survey drone over the campus from a launch pad on the playground. With the pad as origin, x east, y north and z up in metres, every position, flight leg, wind and camera angle becomes a vector, and the chapter follows the drone through a morning's survey. All numbers are made up for the story.

The lesson in notes

In short

A scalar is fixed by one real number (with a unit where needed): length, mass, time, temperature, volume, density. A vector needs a magnitude and a direction: displacement, velocity, acceleration, force.

A vector is drawn as a directed line segment. The vector AB starts at the initial point A and ends at the terminal point B; its magnitude, written |AB|, is the length AB. In print vectors are set in bold (a); by hand an arrow is drawn over the letter.

A length is never negative, so |a| < 0 is meaningless; |a| = 0 only for the zero vector.

Fix an origin O and axes. The position vector of P(x, y, z) is OP = r, with |r| = √(x² + y² + z²). For P(3, 4, 12), |OP| = √(9 + 16 + 144) = 13.

If r 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, and x = lr, y = mr, z = nr. For P(3, 4, 12) they are 3/13, 4/13, 12/13.

Direction cosines always satisfy l² + m² + n² = 1. Any three numbers a, b, c proportional to them (a = λl, b = λm, c = λn) are direction ratios, and a² + b² + c² need not be 1.

Example: a displacement of 40 km, 30° west of south, is drawn as a 40-unit arrow turned 30° from due south towards the west. Both the 40 and the direction are part of the answer.