# Math Insight

### Dot product examples

Given the geometric definition of the dot product along with the dot product formula in terms of components, we are ready to calculate the dot product of any pair of two- or three-dimensional vectors.

#### Example 1

Calculate the dot product of $\vc{a}=(1,2,3)$ and $\vc{b}=(4,-5,6)$. Do the vectors form an acute angle, right angle, or obtuse angle?

Solution: Using the component formula for the dot product of three-dimensional vectors, \begin{gather*} \vc{a} \cdot \vc{b} = a_1b_1+a_2b_2+a_3b_3, \end{gather*} we calculate the dot product to be $$\vc{a} \cdot \vc{b} = 1(4)+2(-5)+3(6) = 4-10+18=12.$$

Since $\vc{a} \cdot \vc{b}$ is positive, we can infer from the geometric definition, that the vectors form an acute angle.

#### Example 2

Calculate the dot product of $\vc{c}=(-4,-9)$ and $\vc{d}=(-1,2)$. Do the vectors form an acute angle, right angle, or obtuse angle?

Solution: Using the component formula for the dot product of two-dimensional vectors, \begin{gather*} \vc{a} \cdot \vc{b} = a_1b_1+a_2b_2, \end{gather*} we calculate the dot product to be $$\vc{c}\cdot \vc{d} = -4(-1)-9(2) = 4-18=-14.$$

Since $\vc{c} \cdot \vc{d}$ is negative, we can infer from the geometric definition, that the vectors form an obtuse angle.

#### Example 3

If $\vc{a}=(6,-1,3)$, for what value of $c$ is the vector $\vc{b}=(4,c,-2)$ perpendicular to $\vc{a}$?

Solution: For $\vc{a}$ and $\vc{b}$ to be perpendicular, we need their dot product to be zero. Since $$\vc{a}\cdot\vc{b} = 6(4)-1(c)+3(-2) = 24-c-6=18-c,$$ the number $c$ must satisfy $18-c=0$, or $c=18$.

You can double-check that the vector $\vc{b}=(4,18,-2)$ is indeed perpendicular to $\vc{a}$ by verifying that $\vc{a}\cdot\vc{b} = (6,-1,3)\cdot(4,18,-2) =0$.