The dot product combines two vectors and creates a scalar that gives us geometric information about the input vectors. If both vectors are the same, then \(\vec{v} \cdot \vec{v}\) gives us the square of the length of \(\vec{v}\text{.}\) The length of a vector \(\vec v\) in \(\IR^n\text{,}\) denoted \(\lvert \vec v \rvert\text{,}\) is defined as
Consider each of the following properties of the dot product. Label each property as valid if the property holds for Euclidean vectors \(\vec u\text{,}\)\(\vec v\) and \(\vec w\) from \(\IR^n\text{,}\) and scalars \(a,b \in \IR\text{,}\) and invalid if it does not.
\(\left(\vec u \cdot \vec v\right) \cdot \vec w=\vec u \cdot \left(\vec v \cdot \vec w\right)\text{.}\)
Like arithmetic of real numbers, the dot product on vectors satisfies some familiar properties. Let \(\vec u\text{,}\)\(\vec v\) and \(\vec w\) be vectors from \(\IR^n\text{,}\) and let \(a,b \in \IR\) be scalars.
\(\vec u \cdot \vec v = \vec v \cdot \vec u\text{.}\)
\(\left( a\vec u\right) \cdot \vec v = a\left(\vec u \cdot \vec v\right)\text{.}\)
\(\left(a\vec u + b \vec v\right)\cdot \vec w =a \vec u \cdot \vec w + b \vec v \cdot \vec w\text{.}\)
Activity6.1.5.
Given the linear transformation \(S:\IR^2 \to \IR^2\) whose standard matrix is \(\left[\begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array}\right]\) and vector \(\vec v = \left[\begin{array}{c} 3 \\ 4 \end{array}\right]\text{,}\)
(a)
Graph \(\vec v\) and \(S( \vec v )\text{.}\)
(b)
For an unspecified vector \(\vec w = \left[\begin{array}{c} w_1 \\ w_2 \end{array}\right]\text{,}\) describe the relationship between \(\vec w\) and \(S( \vec w )\text{.}\)
Activity6.1.6.
Consider \(\vec v = \left[\begin{array}{c} 3 \\ 4 \end{array}\right]\text{.}\)
(a)
What vector \(\vec w = \left[\begin{array}{c} ? \\ ? \end{array}\right]\) is the result of rotating \(\vec v\) by \(90^{\circ}\) counter-clockwise?
(b)
Find the dot product \(\vec v \cdot \vec w\text{.}\)
(c)
For an arbitrary vector \(\vec x = \left[\begin{array}{c} x_1 \\ x_2 \end{array}\right]\text{,}\) what vector \(\vec y = \left[\begin{array}{c} ? \\ ? \end{array}\right]\) is the result of rotating \(\vec x\) by \(90^{\circ}\) counter-clockwise?
(d)
Find the dot product \(\vec x \cdot \vec y\text{.}\)
(e)
Suppose two vectors are perpendicular. What can you say about their dot product?
Definition6.1.7.
Two vectors \(\vec u\) and \(\vec v\) in \(\IR^n\) are orthogonal provided \(\vec u \cdot \vec v = 0\text{.}\)
Definition6.1.8.
Given two vectors \(\vec u\) and \(\vec v\) in \(\IR^n\text{,}\) the distance between the vectors, denoted \(d(\vec u,\vec v)\) is given by