# Geometric vectors

A vector describes a movement from one point to another.

## Vector notation

A vector quantity has both direction and .

(In contrast a scalar quantity has magnitude only - eg, the numbers 1, 2, 3, 4...)

The diagram above represents a vector. The arrow displays its direction, hence this vector can be written as $$\overrightarrow {AB}$$, a, or $$\begin{pmatrix} 3 \\ 4 \end{pmatrix}$$.

In print, a is written in bold type. In handwriting, the vector is indicated by underlining the letter.

If we reverse the arrow it now points from B to A.

Remember that the arrow describes the direction. So, in this case, the vector is from B to A.

If we move 'backwards' along a vector, it becomes negative, so a becomes -a. Moving from B to A entails moving 3 units to the left, and 4 down.

So the three ways to write this vector are: $$\overrightarrow {BA}$$, -a and $$\begin{pmatrix} -3 \\ -4 \end{pmatrix}$$.

## Equal vectors

If two vectors have the same magnitude and direction, then they are equal regardless of their position.

$\left( \begin{array}{l} a\\ b \end{array} \right) + \left( \begin{array}{l} c\\ d \end{array} \right) = \left( \begin{array}{l} a + c\\ b + d \end{array} \right)$

Look at the graph below to see the movements between PQ, QR and PR.

Vector $$\overrightarrow {PQ}$$ followed by vector $$\overrightarrow {QR}$$ represents a movement from P to R.

$\overrightarrow {PQ} + \overrightarrow {QR} = \overrightarrow {PR}$

Written out the vector addition looks like this:

$\left( \begin{array}{l}2\\5\end{array} \right) + \left( \begin{array}{l}\,\,\,\,\,4\\- 3\end{array} \right) = \left( \begin{array}{l}6\\2\end{array} \right)$

## Subtracting vectors

Subtracting a vector is the same as adding a negative version of the vector (remember that making a vector negative means reversing its direction).

$\left( \begin{array}{l} a\\ b \end{array} \right) - \left( \begin{array}{l} c\\ d \end{array} \right) = \left( \begin{array}{l} a - c\\ b - d \end{array} \right)$

Look at the diagram and imagine going from X to Z. How would you write the path in vectors using only the vectors $$\overrightarrow {XY}$$ and $$\overrightarrow {ZY}$$?

You could say it is vector $$\overrightarrow {XY}$$ followed by a backwards movement along $$\overrightarrow {ZY}$$.

So we can write the path from X to Z as:

$\overrightarrow {XY} - \overrightarrow {ZY} = \overrightarrow {XZ}$

Written out in numbers it looks like this:

$\left( \begin{array}{l} 4\\ 2 \end{array} \right) - \left( \begin{array}{l} 1\\ 2 \end{array} \right) = \left( \begin{array}{l} 3\\ 0 \end{array} \right)$

Question

If $$x = \left( \begin{array}{l} 1\\ 3 \end{array} \right)$$, $$y = \left( \begin{array}{l} - 2\\ 4 \end{array} \right)$$ and $$z = \left( \begin{array}{l} - 1\\ - 2 \end{array} \right)$$ find:

1. $- y$
2. $x - y$
3. $2x + 3z$

1. $$\left( \begin{array}{l}\,\,\,\,\,2\\- 4\end{array} \right)$$ Did you remember to change the signs?
2. $\left( \begin{array}{l}\,1\\3\end{array} \right) - \left( \begin{array}{l}- 2\\\,\,\,\,4\end{array} \right) = \left( \begin{array}{l}\,1 - - 2\\3 - \,\,\,\,4\end{array} \right) = \left( \begin{array}{l}\,\,\,\,\,3\\- 1\end{array} \right)$
3. $\left( \begin{array}{l}\,1\\3\end{array} \right) + 3\left( \begin{array}{l}- 1\\- 2\end{array} \right) = \left( \begin{array}{l}2\\6\end{array} \right) + \left( \begin{array}{l}- 3\\- 6\end{array} \right) = \left( \begin{array}{l}- 1\\\,\,\,\,0\end{array}\right)$

## Resultant vectors

A resultant vector is a vector that 'results' from adding two or more vectors together.

To travel from X to Z, it is possible to move along vector $$\overrightarrow {XY}$$ followed by $$\overrightarrow {YZ}$$. It is also possible to go directly along $$\overrightarrow {XZ}$$.

$$\overrightarrow {XZ}$$ is therefore known as the resultant of $$\overrightarrow {XY}$$ and $$\overrightarrow {YZ}$$ .

Question

Write as single vectors:

1.$$f + g$$

2.$$a + b$$

3.$$e - b - a$$

1.$$e$$

2.$$- c$$ (Did you remember the minus sign?)

3.$$- d$$

Remember: Two vectors are equal if they have the same magnitude and direction, regardless of where they are on the page.

Question

Triangles ABC and XYZ are equilateral.

X is the midpoint of AB, Y is the midpoint of BC, Z is the midpoint of AC.

$$\overrightarrow {AX} = a$$, $$\overrightarrow {XZ} = b$$, $$\overrightarrow {AZ} = c$$

Express each of the following in terms of a, b and c.

1. $\overrightarrow {XY}$
2. $\overrightarrow {YZ}$
3. $\overrightarrow {XC}$
4. $\overrightarrow {BZ}$
5. $\overrightarrow {AC}$

1. c
2. - a Remember that $$\overrightarrow {YZ}$$ is to $$\overrightarrow {AX}$$ and of the same length, but the direction is different.
3. b + c (It is also possible to move from X to A and then on to C. This would give the answer - a + 2c. How many other answers can you think of?)
4. b - a or 2b - c or - 2a + c
5. 2c