Lesson 1 of 10 · 6 min

The scalar product

NCERT § "The Scalar Product"

Kiran pulls the supply trolley with a rope at 60° to the ground. He pulls with 50 N, yet the trolley only feels part of that along the road. Which part, and how do we multiply a force by a distance when they point different ways?

The story this chapter follows: Setting up the temple fair

Anu and her brother Kiran spend the day before the village temple fair setting up their family's stall. They drag a 40 kg trolley of supplies, shove crates across the stage, dodge a falling coconut, test a dart-gun game, run the pump that fills the water tank, and finally ride the bumper cars. Every idea in this chapter is somewhere in that day. Take g = 10 m/s² in the story (the sims use 9.8).

The lesson in notes

In short

The scalar (dot) product of two vectors is A·B = AB cos θ, with θ the angle between them; the result is a scalar.

In components, A·B = AxBx + AyBy + AzBz; for unit vectors î·î = 1 and î·ĵ = 0.

The dot product is commutative (A·B = B·A) and distributive over addition.

A·B is positive for θ < 90°, zero for θ = 90° and negative for θ > 90°.

Geometrically, A·B is the magnitude of one vector times the projection of the other on it.

The angle between two vectors can be found from cos θ = A·B/(AB).