Vectors

A vector describes a movement from one point to another. A vector quantity has both direction and magnitude (size).

A scalar quantity has only magnitude.

A vector can be represented by a line segment labelled with an arrow.

Vector lines

A vector between two points A and B is described as: \overrightarrow{AB}, \mathbf{a} or \underline{a}.

Vector AB

The vector can also be represented by the column vector \begin{pmatrix} 3 \\ 4 \end{pmatrix}. The top number is how many to move in the positive x-direction and the bottom number is how many to move in the positive y-direction.

Vectors are equal if they have the same magnitude and direction regardless of where they are.

\overrightarrow{CD} = \begin{pmatrix} 1 \\ 4 \end{pmatrix}

\overrightarrow{EF} = \begin{pmatrix} 1 \\ 4 \end{pmatrix}

So \overrightarrow{CD} = \overrightarrow{EF}

Vectors CD and EF

A negative vector has the same magnitude but the opposite direction.

Vectors k and -k

Vector \mathbf{-k} is the same as travelling backwards down the vector \mathbf{k}.

Example

Write, in terms of \mathbf{a}, \mathbf{b} and \mathbf{c}, the vectors \overrightarrow{ZY}, \overrightarrow{YC}, \overrightarrow{ZA} and \overrightarrow{BX}.

Vectors AX, XB, BY, YC, CZ, ZA, XZ, XY and ZY (triangle of 4 triangles)

\overrightarrow{ZY} = \mathbf{a}

\overrightarrow{ZY} and \overrightarrow{AX} are equal vectors, they have the same magnitude and direction.

\overrightarrow{YC} = \mathbf{b}

\overrightarrow{YC} and \overrightarrow{XZ} are equal vectors, they have the same magnitude and direction.

\overrightarrow{ZA} = \mathbf{-c}

\overrightarrow{ZA} has the same magnitude as \overrightarrow{AZ} but the opposite direction.

\overrightarrow{BX} = \mathbf{-a}

\overrightarrow{BX} has the same magnitude as \overrightarrow{AX} but the opposite direction.