Position Vectors (Edexcel IAL Maths)

Spec point — 7.4

Position vectors

What is a position vector?

• Position vectors describe the position of a point in relation to the origin
• They are different to displacement vectors which describe the direction and distance between any two points

Distance between two points

• The distance between two points is the magnitude of the vector between them (see Magnitude Direction)

How do I find the magnitude of a displacement vector?

• You can use coordinate geometry to find magnitudes of displacement vectors from A to B

  • From the position vectors of A and B you know their coordinates

    • If $a = \vec{OA} = x_{1} i + y_{1} j = ( \begin{gathered} x_{1} \\ y_{1} \end{gathered} )$, then point A has coordinates $( x_{1} , y_{1} )$
    • If $b = \vec{OB} = x_{2} i + y_{2} j = ( \begin{gathered} x_{2} \\ y_{2} \end{gathered} )$, then point B has coordinates $( x_{2} , y_{2} )$

  • The distance between two points is given by $d = \sqrt{( x_{1} - x_{2} )^{2} + ( y_{1} - y_{2} )^{2}}$

    • So $\left| \vec{AB} \right| = \sqrt{( x_{1} - x_{2} )^{2} + ( y_{1} - y_{2} )^{2}}$

  • For example, if points A and B have position vectors $5i+3j$ and $3i-6j$ respectively

    • then $\left| \vec{AB} \right| = \sqrt{( 5 - 3 )^{2} + ( 3 - ( - 6 ) )^{2}} = \sqrt{85} = 9.22 ( 3 \mathrm{s} . \mathrm{f} . )$

• Alternatively, you could find $\left| \vec{AB} \right|$ by

  • first using $\overset{\rightarrow}{AB}=-\overset{\rightarrow}{OA}+\overset{\rightarrow}{OB}$ to find $\overset{\rightarrow}{AB}$ in vector form

    • and then calculating its magnitude directly

  • See the Worked Example below

Exam Hint

• Remember if asked for a position vector, you must find the vector all the way from the origin.
• Diagrams can help, if there isn’t one, draw one.

Worked Example