Scalars, vectors and adding them

A scalar has only a size: mass, time, distance, speed, work. A vector has a size and a direction, and it adds by the triangle law: displacement, velocity, acceleration, force.

Having a direction is not enough. Electric current flows one way along a wire, but two currents meeting at a junction simply add as numbers, so current is a scalar.

Some kinds of vectors you will meet:

• Equal vectors: the same size and the same direction.
• A negative vector, –A: the same size as A, the opposite direction.
• The zero (null) vector: size zero, as in A – A = 0.
• A unit vector: size 1, used only to show a direction. Â = A/|A|, and î, ĵ, k̂ point along x, y and z.

Triangle law: put the tail of B on the head of A. The arrow from the start of A to the end of B is A + B. Walk 5 m east and then 12 m north: you end 13 m from where you started.

Parallelogram law: put the two tails together and complete the parallelogram; its diagonal from the shared tail is A + B. Both laws give the same answer, and the order does not matter: A + B = B + A.

For many vectors, keep going head to tail (the polygon law). If the last head comes back to the first tail, the loop is closed and the vectors add up to zero: three forces that form a closed triangle balance each other.

To subtract, add the negative: A – B = A + (–B). Multiplying by a number λ stretches the size by |λ|, and a negative λ also turns the arrow round.

The resultant can never be longer than the two sizes put together: |A + B| ≤ |A| + |B|, with equal only when A and B point the same way. 3 N and 4 N can give anything from 1 N (opposite) to 7 N (same way).