Position & Displacement Vectors (Cambridge International IGCSE Mathematics)

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 $( \begin{gathered} 3 \\ - 2 \end{gathered} )$ from the origin

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 $( \begin{gathered} 3 \\ 2 \end{gathered} )$ and $( \begin{gathered} 6 \\ - 10 \end{gathered} )$respectively.

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

Answer:

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 $q$* *

$\vec{PQ} = ( \begin{gathered} 6 \\ - 10 \end{gathered} ) - ( \begin{gathered} 3 \\ 2 \end{gathered} )$

Expand and simplify

$\begin{gathered} = ( \begin{gathered} 6 - 3 \\ - 10 - 2 \end{gathered} ) \\ = ( \begin{gathered} 3 \\ - 12 \end{gathered} ) \end{gathered}$

P Q -vec = stretchy ( | 3 | - 12 | | stretchy )