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Vectors - EdexcelVectors

A vector quantity has both size and direction. Vectors can be added, subtracted and multiplied by a scalar. Geometrical problems can be solved using vectors.

Part ofMathsGeometry and measure

Vectors

Click to explore updated revision resources for GCSE Maths: Vectors, with step-by-step slideshows, quizzes, practice exam questions, and more!

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

A 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

Triangle ABC is isosceles. X is the midpoint of AB, Y is the midpoint of BC and Z is the midpoint of AC.

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.

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Vector arithmetic

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