Position & Displacement Vectors (DP IB Maths: AA HL)

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Adding & Subtracting Vectors

How are vectors added and subtracted numerically?

To add or subtract vectors numerically simply add or subtract each of the corresponding components
In column vector notation just add the top, middle and bottom parts together

For example: $(\binom{2}{\begin{matrix} 1 \\ - 5 \end{matrix}}) - (\binom{1}{\begin{matrix} 4 \\ 3 \end{matrix}}) = (\binom{1}{\begin{matrix} - 3 \\ - 8 \end{matrix}})$

In base vector notation add each of the i, j, and k components together separately

For example: (2i + j – 5k) – (i + 4j + 3k) = (i – 3j – 8k)

Vector Addition Diagram 1b

How are vectors added and subtracted geometrically?

Vectors can be added geometrically by joining the end of one vector to the start of the next one
The resultant vector will be the shortest route from the start of the first vector to the end of the second

A resultant vector is a vector that results from adding or subtracting two or more vectors

If the two vectors have the same starting position, the second vector can be translated to the end of the first vector to find the resultant vector

This results in a parallelogram with the resultant vector as the diagonal

To subtract vectors, consider this as adding on the negative vector

For example: a – b = a + (-b)
The end of the resultant vector a – b will not be anywhere near the end of the vector b

Instead, it will be at the point where the end of the vector -b would be

Exam Tip

Working in column vectors tends to be easiest when adding and subtracting
- in your exam, it can help to convert any vectors into column vectors before carrying out calculations with them
If there is no diagram, drawing one can be helpful to help you visualise the problem

Worked example

Find the resultant of the vectors a = 5i – 2j and b = $(\binom{- 3}{\begin{matrix} 1 \\ 2 \end{matrix}})$ .

3-9-2-ib-aa-hl-add--sub-vectors-we-solution-a-

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Position Vectors

What is a position vector?

A position vector describes the position of a point in relation to the origin

It describes the direction and the distance from the point O: 0i + 0j + 0k or $(\binom{0}{\begin{matrix} 0 \\ 0 \end{matrix}})$
It is different to a displacement vector which describes the direction and distance between any two points

The position vector of point A is written with the notation a = $\vec{OA}$

The origin is always denoted O

The individual components of a position vector are the coordinates of its end point

For example the point with coordinates (3, -2, -1) has position vector 3i – 2j – k

Worked example

Determine the position vector of the point with coordinates (4, -1, 8).

3-9-2-ib-aa-hl-position-vector-we-solution

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Displacement Vectors

What is a displacement vector?

A displacement vector describes the shortest route between any two points
- It describes the direction and the distance between any two points
- It is different to a position vector which describes the direction and distance from the point O: 0i + 0j or $(\binom{0}{0})$
The displacement vector of point B from the point A is written with the notation $\vec{AB}$
A displacement vector between two points can be written in terms of the displacement vectors of a third point
- $\vec{AB} = \vec{AC} + \vec{CB}$
A displacement vector can be written in terms of its position vectors
- For example the displacement vector $\vec{AB}$ can be written in terms of $\vec{OA}$ and $\vec{OB}$
- $\vec{AB} = \vec{AO} + \vec{OB} = - \vec{OA} + \vec{OB} = \vec{OB} - \vec{OA}$
- For position vector a = $\vec{OA}$ and b = $\vec{OB}$ the displacement vector $\vec{AB}$ can be written b – a

Exam Tip

In an exam, sketching a quick diagram can help to make working out a displacement vector easier

Worked example

The point A has coordinates (3, 0, -1) and the point B has coordinates (-2, -5, 7). Find the displacement vector $\vec{AB}$ .

3-9-2-vectors-we-solution-position-and-displacement-wqe-solution

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Position & Displacement Vectors (DP IB Maths: AA HL)

Revision Note

How are vectors added and subtracted numerically?

How are vectors added and subtracted geometrically?

What is a position vector?

What is a displacement vector?

Author: Amber