Describe a move in the plane

A vector records magnitude and direction. Subtract position vectors to find the displacement from one point to another. Adding vectors joins displacements head to tail.

The magnitude comes from Pythagoras. Divide a nonzero vector by its magnitude to obtain a unit vector; multiplying by a negative scalar reverses its direction.

\displaystyle \overrightarrow{AB}=\mathbf b-\mathbf a,\quad |(x,y)|=\sqrt{x^2+y^2}

WORKED EXAMPLE

Find the displacement from A(−1, 2) to B(5, 10).

Subtract A from B to get (6, 8).
Its magnitude is √(36 + 64) = 10.
A unit vector in this direction is ( $\frac{3}{5}$ , $\frac{4}{5}$ ).

Assumed knowledge

Plot Cartesian coordinates, subtract signed numbers, and use Pythagoras’ theorem to find distances. Recall right-triangle trigonometry and the meaning of magnitude and direction.

Study Specialist Mathematics alongside, or after, Mathematical Methods. Later units build on earlier units.

QCAA prior-learning guidance ↗

Describe a move in the plane

Find the displacement from A(−1, 2) to B(5, 10).

Put the idea to work.