Revision sheet

Vectors and components

• $\overrightarrow{PQ} = \langle x_2 - x_1,\ y_2 - y_1 \rangle$: end minus start.
• Same components, same vector, wherever it starts.

Watch out: start minus end gives the opposite vector.

Vectors in context

1. Write each velocity or force in components.
2. Add them: the resultant.
3. Magnitude = speed or total force; $\tan\theta$ = direction.

Watch out: speeds in different directions do not simply add.

Matrices

Size is rows × columns; add entry by entry; a scalar multiplies every entry.

Watch out: rows first.

Multiplying matrices

(m × n)(n × p) = m × p; entry = row · column.

Watch out: AB is usually not BA.

Matrices as transformations

Av moves vectors; columns are the images of (1, 0) and (0, 1); areas scale by |det|.

Watch out: the area factor is the determinant, not a sum of entries.