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Vectors I

Position vectors and displacement vectors

Position vectors and displacement vectors

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Summary

​Position vectors and displacement vectors

​​In a nutshell

Vectors are used to describe a position of a point in space. A position vector gives the position of a point relative to a fixed origin, OOO. A displacement vector describes movement from one position to another.



Position vectors

A position vector gives the position of a point relative to a fixed origin.

Maths; Vectors I; KS5 Year 12; Position vectors and displacement vectors

Point PPP has coordinates (a,b)(a,b)(a,b). The position vector of PPP from the fixed origin OOO to PPP, is OP→\overrightarrow{OP}OP.​

​OP→=ai+bj=(ab)\boxed{\overrightarrow{OP}=a\textbf{i}+b\textbf{j}=\begin{pmatrix}a\\b\end{pmatrix}}OP=ai+bj=(ab​)​​​



Displacement vectors

A displacement vector describes movement from one position to another. In a two-coordinate system, the difference between the position vectors of the final point and the initial point, relative to the origin, gives your displacement.


Maths; Vectors I; KS5 Year 12; Position vectors and displacement vectors


If point AAA is the initial position, and point BBB is the final position, the displacement vector AB→\overrightarrow{AB}AB is the difference between the two points.


The position vector of AAA is OA→=a\overrightarrow{OA}=\textbf{a}OA=a.

The position vector of BBB is OB→=b\overrightarrow{OB}=\textbf{b}OB=b.


AB→=OB→−OA→\boxed{\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}}AB=OB−OA​​​



Example 1

The diagram shows points AAA and BBB. Find, in column vector form, the position vector of AAA, the position vector of BBB, and the vector AB→\overrightarrow{AB}AB.

Maths; Vectors I; KS5 Year 12; Position vectors and displacement vectors


The position vector of AAA is its position relative to the origin, OOO:

​OA→=(−12)‾\underline{\overrightarrow{OA}=\begin{pmatrix}-1\\2\end{pmatrix}}OA=(−12​)​​​


The position vector of BBB is its positio​n relative to the origin, OOO:

​OB→=(52)‾\underline{\overrightarrow{OB}=\begin{pmatrix}5\\2\end{pmatrix}}OB=(52​)​​​


AB→\overrightarrow{AB}AB is the displacement vector between t​he two points:

​AB→=OB→−OA→=(52)−(−12)=(60)‾\begin{aligned}\overrightarrow{AB}&=\overrightarrow{OB}-\overrightarrow{OA}\\&=\begin{pmatrix}5\\2\end{pmatrix}-\begin{pmatrix}-1\\2\\\end{pmatrix}\\&=\underline{\begin{pmatrix}6\\0\end{pmatrix}}\end{aligned}AB​=OB−OA=(52​)−(−12​)=(60​)​​​​


Example 2

Points AAA and BBB have position vectors 4i+3j4\textbf{i}+3\textbf{j}4i+3j and −3i+j-3\textbf{i}+\textbf{j}−3i+j respectively. Find AB→\overrightarrow{AB}AB.


AB→=OB→−OA→=(−3i+j)−(4i+3j)=−7i−2j‾\begin{aligned}\overrightarrow{AB}&=\overrightarrow{OB}-\overrightarrow{OA}\\&=(-3\textbf{i}+\textbf{j})-(4\textbf{i}+3\textbf{j})\\&=\underline{-7\textbf{i}-2\textbf{j}}\end{aligned}AB​=OB−OA=(−3i+j)−(4i+3j)=−7i−2j​​


Example 3

Point AAA has coordinates (3,2)(3,2)(3,2) and AB→=−2i−8j\overrightarrow{AB}=-2\textbf{i}-8\textbf{j}AB=−2i−8j. Find the position vector of BBB and its exact magnitude.


The position vector of AAA is:

OA→=3i+2j\overrightarrow{OA}=3\textbf{i}+2\textbf{j}OA=3i+2j


We know that AB→=OB→−OA→\begin{aligned}\overrightarrow{AB}&=\overrightarrow{OB}-\overrightarrow{OA}\end{aligned}AB​=OB−OA​. So:

AB→=OB→−OA→OB→=AB→+OA→=(−2i−8j)+(3i+2j)=i−6j‾\begin{aligned}\overrightarrow{AB}&=\overrightarrow{OB}-\overrightarrow{OA}\\\overrightarrow{OB}&=\overrightarrow{AB}+\overrightarrow{OA}\\&=(-2\textbf{i}-8\textbf{j})+(3\textbf{i}+2\textbf{j})\\&=\underline{\textbf{i}-6\textbf{j}}\end{aligned}ABOB​=OB−OA=AB+OA=(−2i−8j)+(3i+2j)=i−6j​​​​

​​

For the magnitude of the position vector of BBB:

∣OB→∣=12+(−6)2=1+36=37‾\begin{aligned}|\overrightarrow{OB}|&=\sqrt{1^2+(-6)^2}\\&=\sqrt{1+36}\\&=\underline{\sqrt{37}}\end{aligned}∣OB∣​=12+(−6)2​=1+36​=37​​​​​


Note: You have been asked for the exact value, so leave your answer in surd form.


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FAQs - Frequently Asked Questions

What is a position vector?

A position vector gives the position of a point relative to a fixed origin.

What is a displacement vector?

A displacement vector describes movement from one position to another.

How do you find the displacement vector between two points?

If point A is the initial position, and point B is the final position, the displacement vector is the difference between the position vectors of the two points.

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