The Scalar (Dot) Product (Cambridge (CIE) A Level Maths: Pure 3): Revision Note

The scalar (dot) product

The scalar product is an important link between the algebra of vectors and the trigonometry of vectors. We shall see that the scalar product is somewhat comparable to the operation of multiplication on real numbers.

What is the scalar (dot) product?

• The scalar product between two vectors a and b is represented by $\mathbf{a}·\mathbf{b}$

  • This is also called the dot product because of the symbol used

• The scalar product between two vectors $\mathbf{a}=a_1\mathbf{i}+a_2\mathbf{j}+a_3\mathbf{k}$ and $\mathbf{b}=b_1\mathbf{i}+b_2\mathbf{j}+b_3\mathbf{k}$ is defined as $\mathbf{a}·\mathbf{b}={\mathrm{a}}_1{b}_1+a_2{b}_2+a_3{b}_3$

• The result of taking the scalar product of two vectors is a real number

  • i.e. a scalar

• For example,

$\left(3\mathbf{i}-\mathbf{k}\right)·\left(2\mathbf{i}+9\mathbf{j}+\mathbf{k}\right)=3\times 2+0\times 9+\left(-1\right)\times 1=6+0-1=5$

and

$\left(\begin{array}{c}2\\ 7\end{array}\right)·\left(\begin{array}{c}-8\\ 2\end{array}\right)=2\times \left(-8\right)+7\times 2=-16+14=-2$

• The scalar product has some important properties:

  • The order of the vectors doesn’t affect the result:

$\mathbf{a}·\mathbf{b}=\mathbf{b}·\mathbf{a}$

• In effect we can ‘multiply out’ brackets:

$\mathbf{a}\mathbf{·}\mathbf{(}\mathbf{b}\mathbf{+}\mathbf{c}\mathbf{)}\mathbf{=}\mathbf{a}\mathbf{·}\mathbf{b}\mathbf{+}\mathbf{a}\mathbf{·}\mathbf{c}$

• This means that we can do many of the same things with vectors as we can do when operating on real numbers – for example,

$\left(\mathbf{a}-\mathbf{b}\right)·\left(\mathbf{a}-\mathbf{b}\right)=\mathbf{a}\mathbf{·}\mathbf{a}-2\mathbf{a}\mathbf{·}\mathbf{b}+\mathbf{b}\mathbf{·}\mathbf{b}$

• The scalar product between a vector and itself is equal to the square of its magnitude:

$\mathbf{a}\mathbf{·}\mathbf{a}={\left|\mathbf{a}\right|}^2$

For example,

 $\left(\begin{array}{c}2\\ 7\end{array}\right)·\left(\begin{array}{c}2\\ 7\end{array}\right)=2^2+7^2=53$  and  ${\left|\left(\begin{array}{c}2\\ 7\end{array}\right)\right|}^2=2^2+7^2=53$

What is the formula for the scalar product?

• There is another important method for finding $\mathbf{a}\mathbf{·}\mathbf{b}$involving the angle between the two vectors $\theta$:

$\mathbf{a}\mathbf{·}\mathbf{b}=\left|\mathbf{a}\right|\left|\mathbf{b}\right|\cos \theta$

• Here $\theta$ is the angle between the vectors when they are placed ‘base to base’

  • when the vectors are placed so that they begin at the same point

• This formula can be derived using the cosine rule and expanding $\left(\mathbf{a}-\mathbf{b}\right)·\left(\mathbf{a}-\mathbf{b}\right)$