ObjectivesTopical
perform the dot-product of two vectors
recognize when two vectors are orthogonal
Section2.2The Dot Product
¶Now that we've learned how to add and subtract vectors, stretch them by scalars, and use them to find lines, it's time to introduce a way of multiplying vectors called the dot product. We'll use the dot product to help us find find angles. First, we need to recall the law of cosines.
Theorem2.2.1The Law of Cosines
Consider a triangle with side lengths \(a\text{,}\) \(b\text{,}\) and \(c\text{.}\) Let \(\theta\) be the angle between the sides of length \(a\) and \(b\text{.}\) Then the law of cosines states that
\begin{equation*} c^2=a^2+b^2-2ab\cos\theta. \end{equation*}If \(\theta=90^\circ\text{,}\) then \(\cos\theta=0\) and this reduces to the Pythagorean theorem.
Exercise2.2.1
Sketch in \(\mathbb{R}^2\) the vectors \(\left\lt 1,2\right>\) and \(\left\lt 3,5\right>\text{.}\) Use the law of cosines to find the angle between the vectors.
Exercise2.2.2
Sketch in \(\mathbb{R}^3\) the vectors \(\left\lt 1,2,3\right>\) and \(\left\lt -2,1,0\right>\text{.}\) Use the law of cosines to find the angle between the vectors.
Subsection2.2.1Introducing the Dot Product
Definition2.2.2The Dot Product
If \(\vec u = (u_1,u_2,u_3)\) and \(\vec v= (v_1,v_2,v_3)\) are vectors in \(\mathbb{R}^3\text{,}\) then we define the dot product of these two vectors to be
\begin{equation*} \vec u\cdot \vec v = u_1 v_1+ u_2 v_2+ u_3 v_3. \end{equation*}A similar definition holds for vectors in \(\mathbb{R}^n\text{,}\) where \(\vec u\cdot \vec v = u_1 v_1+ u_2 v_2+\cdots+ u_n v_n.\) You just multiply corresponding components together and then add. It is the same process used in matrix multiplication.
Exercise2.2.3
(a)
Use the formula \(\vec u\cdot \vec v=|\vec u||\vec v|\cos\theta\) to find the angle between the vectors \(\left\lt 1,2,3\right>\) and \(\left\lt -2,1,0\right>\text{.}\)
(b)
Which was easier, Exercise 2 or this method? (You will derive this formula in a later problem)
Definition2.2.3
We say that the vectors \(\vec u\) and \(\vec v\) are orthogonal if \(\vec u\cdot \vec v=0\text{.}\)
Exercise2.2.4
Find two vectors orthogonal to \((1,2)\text{.}\) Then find 4 vectors orthogonal to \((3,2,1)\text{.}\)
Subsection2.2.2Using the Dot Product
Exercise2.2.5
Mark each statement true or false. Use Definitions 2.1.3 - Definition 3 to explain and justify or prove your claim. You can assume that \(\vec u,\vec v,\vec w\in\mathbb{R}^3\) and that \(c\in\mathbb{R}\text{.}\)
(a)
\(\vec u\cdot \vec v=\vec v\cdot \vec u\text{.}\)
(b)
\(\vec u\cdot (\vec v\cdot \vec w)=(\vec u\cdot\vec v)\cdot\vec w\text{.}\)
(c)
\(c(\vec u\cdot \vec v)=(c\vec u)\cdot \vec v=\vec u\cdot (c\vec v)\text{.}\)
(d)
\(\vec u+(\vec v\cdot \vec w)=(\vec u+\vec v)\cdot(\vec u+\vec w)\text{.}\)
(e)
\(\vec u\cdot (\vec v+ \vec w)=(\vec u\cdot \vec v)+(\vec u\cdot\vec w)\text{.}\)
(f)
\(\vec u\cdot \vec u= |\vec u|^2\text{.}\)
Exercise2.2.6
If \(\vec u = (u_1,u_2,u_3)\) and \(\vec v= (v_1,v_2,v_3)\) are vectors in \(\mathbb{R}^3\) (which is often written \(\vec u,\vec v\in\mathbb{R}^3\)), then show that
\begin{equation*} |\vec u-\vec v|^2 = |\vec u|^2-2\vec u\cdot \vec v +|\vec v|^2. \end{equation*}Exercise2.2.7
Let \(\vec u,\vec v\in\mathbb{R}^3\text{.}\) Let \(\theta\) be the angle between \(\vec u\) and \(\vec v\text{.}\)
(a)
Use the law of cosines to explain why \(|\vec u-\vec v|^2=|\vec u|^2+|\vec v|^2-2|\vec u||\vec v|\cos\theta\text{.}\)
(b)
Use the above together with Exercise 6 to derive
\begin{equation*} \vec u\cdot \vec v=|\vec u||\vec v|\cos\theta. \end{equation*}Exercise2.2.8
Show that if two nonzero vectors \(\vec u\) and \(\vec v\) are orthogonal, then the angle between them is 90\(^\circ\text{.}\) Then show that if the angle between them is 90\(^\circ\text{,}\) then the vectors are orthogonal. I.E. expand and compute both sides of the formula \(\vec u\cdot \vec v=|\vec u||\vec v|\cos\theta\) with non-zero, orthogonal vectors.
The dot product provides a really easy way to find when two vectors meet at a right angle. The dot product is precisely zero when this happens.
Subsection2.2.3Computational Practice
These are provided to help you achieve better skills in basic computational answers