Position vectors

Page 1

What are position vectors?

A position vector is the vector of an arrow drawn to from an origin to a point.

Look at the example below - a Cartesian plane in which we see points AA

A

and BB

B

The position vector of AA

A

connects the origin, marked OO

O

, to AA

A

- it can be written as a⃗ a→

\vec a

or OA→OA→

\vec{OA}

.

Similarly, the position vector of \(B\) connects the origin to BB

B

- it can be written as b⃗ b→

\vec b

or OB→OB→

\vec{OB}

It is important that the letters are written in this order. The vector BO→BO→

\vec{BO}

would connect B to the origin and therefore represents −b⃗ −b→

-\vec b

.

If you are using Cartesian components to describe a position vector, the Cartesian coordinates of a point can be used to describe the position vector of that point.

For example, AA

A

has the coordinates (1,4)(1,4)

(1,4)

. This means its position vector can be written as a⃗ =a→=

\vec a=1ı⃗ +4ȷ⃗ 1ı→+4ȷ→

1 \vec{\imath} + 4 \vec{\jmath}

or a⃗ =[14]a→=[14]

\vec a= \begin{bmatrix}1\\4 \\\end{bmatrix}

.

Similarly, BB

B

has the coordinates (−3,−1)(−3,−1)

(-3,-1)

. This means its position vector can be written as b⃗ =b→=

\vec b=−3ı⃗ +−1ȷ⃗ −3ı→+−1ȷ→

-3 \vec{\imath} + -1 \vec{\jmath}

or b⃗ =[−3−1]b→=[−3−1]

\vec b= \begin{bmatrix}-3\\-1 \\\end{bmatrix}

.

Page 2

Distance between two points

Position vectors can be used to describe the vector between two points.

The position vector \( \vec{AB}\) goes from \(A\) to \(B\)

Going from \(A\) to \(B\) is the same as going from \(A\) to \(O\) and then from \(O\) to \(B\)

Therefore, \( \vec{AB}= \vec b - \vec a\)

This means that the component form of \( \vec{AB}\) can be found by subtracting the components of \(A\) from \(B\), i.e.

\[ \vec{AB}= \vec b - \vec a\]

\[= \begin{bmatrix}-3\\-1 \\\end{bmatrix} -\begin{bmatrix}1\\4 \\\end{bmatrix}=\begin{bmatrix}-4\\-5 \\\end{bmatrix}\] or

\[\Big(-3 \vec{\imath} + -1 \vec{\jmath}\Big) - \Big(1 \vec{\imath} + 4 \vec{\jmath}\Big) = (-3-1) \vec{\imath} + (-1-4) \vec{\jmath}=-4 \vec{\imath}-5 \vec{\jmath}\]

The magnitude of this vector can be used to calculate the distance between the two points. \[ |\vec {AB} | = \sqrt{(-4)^2+(-5)^2}= \sqrt{16+25}=\sqrt{41}\]

Note: \[ \vec{AB}= \vec b - \vec a\]

\[ \vec{BA}= \vec a - \vec b\]

\[ \therefore \vec{BA}=-\vec{AB}\]