Position & Displacement Vectors (Edexcel IGCSE Further Maths)

Spec point — 7D

Position & Displacement Vectors

What are position vectors?

• A position vector describes where a specific point, A, is, relative to a fixed origin, O

  • Lower-case bold (or underlined) letters are used

    • The point A has position vector a = $\overset{\rightarrow}{OA}$

• Their components are equal to their coordinates

  • The point with ****coordinates (3, -2) has position vector 3i – 2j

What are displacement vectors?

• A displacement vector describes the direction and distance between two points

  • The displacement vector from A to B is $\overset{\rightarrow}{AB}$

    • How to get from A to B

• If the points A and B have position vectors a and b relative to O

  • then A to B is the same as A to O (-a) followed by O to B (b)

    • $\overset{\rightarrow}{AB}=-a+b=b-a$
    • This is a useful rule to remember

Exam Hint

• You may need to draw an origin, O, on to a diagram to be able to sketch position vectors.

Worked Example The points $P$ and $Q$ have position vectors 3i + 2j and 6i - 10j respectively.

Find and simplify the vector $\overset{\rightarrow}{PQ}$.

Let p and q be position vectors of P and Q $\overset{\rightarrow}{PQ}$* is the displacement vector from P to Q* Use the rule that $\overset{\rightarrow}{AB}=-a+b=b-a$

$\overset{\rightarrow}{PQ}=q-p$

Substitute in p and q

$\vec{PQ} = ( 6 i - 10 j ) - ( 3 i + 2 j )$

Expand and simplify

$\begin{gathered} = 6 i - 10 j - 3 i - 2 j \\ = 3 i - 12 j \end{gathered}$

$\overset{\rightarrow}{PQ}=3i-12j$