Displacement vectors from points | CAIE 9709 Pure Mathematics 3 Notes

Pure Mathematics 3

Mathematics9709

Add journeys and reverse directions

Add vectors component by component. Geometrically, place the second arrow at the end of the first.
Reversing an arrow changes every sign: $\overrightarrow{BA}=-\overrightarrow{AB}$ .
A closed journey has zero total displacement.

\overrightarrow{AC}=\overrightarrow{AB}+\overrightarrow{BC}

Key idea

Following a complete closed journey returns to the starting point, so the total displacement is the zero vector

\mathbf 0=(0,0,0)

From two points, calculate end minus start

For $A=(x_1,y_1,z_1)$ and $B=(x_2,y_2,z_2)$ , subtract the start coordinates from the end coordinates:

\overrightarrow{AB}=\begin{pmatrix}x_2-x_1\\y_2-y_1\\z_2-z_1\end{pmatrix}

Worked example

A displacement in 3D

Let $A=(2,1,-3)$ and $B=(0,4,1)$ .

\overrightarrow{AB}=\begin{pmatrix}-2\\3\\4\end{pmatrix}

The signs make sense: B is 2 units lower in $x$ , 3 units higher in $y$ and 4 units higher in $z$ .

(Pure Mathematics 2 & 3 Coursebook, Ch. 9.1)

Use signs to check the direction

Common mistake

\overrightarrow{AB}

means B minus A, not A minus B. Say “end minus start” before subtracting.

Your turnIndependent check [3 marks]

Given $B=(0,4,1)$ and $C=(-3,-2,2)$ , find $\overrightarrow{BC}$ and $\overrightarrow{CB}$ . Check their sum.

Show worked answer

\overrightarrow{BC}=\begin{pmatrix}-3\\-6\\1\end{pmatrix}

Answer

\overrightarrow{BC}=(-3,-6,1)

\overrightarrow{CB}=(3,6,-1)

\overrightarrow{BC}+\overrightarrow{CB}=\mathbf 0