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


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Further Mathematics (A2-Level)
Vectors
Question 1 of 7

1. How confident are you in finding

a. the scalar product of two vectors?


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|cos0 where 0 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|cos0 

to give

 cos0=\frac{a.b}{|a||b|} .

In the above example this would give 

\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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