Lesson 1 of 10 · 6 min

Scalars and vectors

NCERT §3.2

Before the match Meera walks from the pavilion 40 m east along the boundary, then 30 m north to her fielding spot. Her fitness band says she walked 70 m. The coach's map says she is 50 m from the pavilion. Both are right.

The story this chapter follows: Meera at the school cricket ground

Meera fields at the boundary of her school's cricket ground. Her walk to her fielding spot, a rain shower in a cross-wind, her throw to the keeper, the camera drone filming the match and her warm-up jog round a painted circle carry every idea of the chapter, with the same numbers throughout.

The lesson in notes

In short

A scalar is fully given by a number with its unit: distance, mass, temperature, time, volume. Scalars combine by ordinary algebra.

A vector has a magnitude and a direction and combines by the triangle law (equivalently the parallelogram law): displacement, velocity, acceleration and force are vectors. In print a vector is bold (v); by hand it is written with an arrow over it.

The position vector r of a point is the arrow from a chosen origin O to that point. If an object moves from P to P′, its displacement is the vector PP′ = r′ − r, which depends only on the start and end points.

The path length between two points is never less than the magnitude of the displacement between them: walking 40 m east and then 30 m north covers 70 m, but the displacement is only 50 m.

A and B are equal vectors when both their sizes and their directions agree. A vector may be shifted parallel to itself without changing it (a free vector); some applications, like a force acting at a point, care about the line of action as well (a localised vector).

The magnitude of a vector is written |A| or simply A; it is never negative.

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Scalars and vectors

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