Preview Activity 9.3.1.
In this activity, we will use the following vectors:
\begin{equation*}
\vu_1=\langle 1,2 \rangle , \vu_2=\langle -1,1 \rangle , \vu_3=\langle 2,-1 \rangle , \vu_4=\langle 1,-2 \rangle
\end{equation*}
(a)
Draw representatives of each of \(\vu_1,\vu_2,\vu_3,\vu_4\) in standard position using the axes below.
A plot of the 2D plane
(b)
Compute each of the dot products listed in the table below. Fill in only the “Value” column at this point. The “Angle Classification” column will be completed in the next part.
| Dot Product | Value | Angle Classification |
|---|---|---|
| \(\vu_1\cdot \vu_2\) | ||
| \(\vu_1\cdot \vu_3\) | ||
| \(\vu_1\cdot \vu_4\) | ||
| \(\vu_2\cdot \vu_4\) | ||
| \(\vu_3\cdot \vu_4\text{.}\) |
(c)
When we look at vectors drawn with the same initial point, as you have done with the vectors in Figure 9.3.1, we can consider the angle between two vectors to be the smaller angle formed by looking at them. This results in an angle between \(0\) and \(\pi\) if we measure in radians or between \(0^{\circ}\) and \(180^{\circ}\) if we measure in degrees. Rather than measuring the angles between the vectors you’ve drawn in Figure 9.3.1 precisely, classify the angle between the pairs of vectors in the table in part b by writing acute, right, or obtuse in the “Angle Classification” column.
(d)
Look at the values of the dot products that you computed in part b and the classifications of the angles between the vectors that you made in part c. Write a sentence or two to describe what you notice about the relationship between the sign of the dot product and the type of angle between the vectors.
(e)
Does the value of the dot product or the classification of the angle change if we change the order in which we consider the vectors?
(f)
Do you think that knowing only the value of the dot product \(\vv\cdot \vw\) would be enough to determine the exact value of the angle between \(\vv\) and \(\vw\) if you didn’t know the components of \(\vv\) and \(\vw\text{?}\) Write a sentence to explain your reasoning.
(g)
Although the angle between a vector and itself is \(0\text{,}\) the dot product of a vector with itself can be computed. Compute \(\vu_1\cdot \vu_1\) and \(\vu_2\cdot \vu_2\) as well as the magnitude of \(\vu_1\) and \(\vu_2\text{.}\) What do you notice about the relationship between these values? Write a sentence explaining why this seems reasonable based on the way the dot product of a vector with itself and the magnitude of that vector are computed.
