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# Vectors : Further Mathematics (A2-Level) : Mathematics Content Knowledge

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Further Mathematics (A2-Level)
Vectors
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# 1. How confident are you in finding

## Example

Find the scalar product of $a.b$ when $a=3i+4j$ and $b=i+2j$.

The formula for the scalar (or dot) product is $a.b=|a||b|cos\theta$ where $\theta$ is the angle between the two vectors. It can also be found by multiplying the $i$ components together and adding the result to the product of the $j$ components (and the product of the $k$ components for three dimensional vectors).

For this example then $a.b=(3x1)+(4x2)=11$

## What this might look like in the classroom

The main use of the scalar product is in finding the angle between two vectors. So re-arrange the formula:

$a.b=|a||b|cos\theta$

to give

$cos\theta =\frac{a.b}{|a||b|}$.

In the above example this would give

$cos\theta =\frac{11}{( \sqrt{3^2+4^2})x(\sqrt{1^2+2^2}) } =\frac{11}{\sqrt{125}}$

which gives

$\theta =10.3^o$.

Students could plot the two vectors on standard coordinate axes and estimate the angle before calculating it.

Students should also consider the implications in obtaining a zero scalar product from two non-zero vectors. This means that the two vectors are parallel (since a zero scalar product would mean the angle between the two vectors has to be zero also).

## Taking this mathematics further

Students could also consider some of the properties of the scalar product. Is it commutatitve – so is a.b the same as b.a? What happens to the scalar product if one of the vectors is doubled?

## Making connections

Students will need to remember how to find the modulus of a vector – this should remind them of their previous study of Pythagoras’ theorem.

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