Theorem 1.5: The Triangle Inequality:
For all $\vu$ and $\vv$ in $\R^n$,
$$
\| \vu + \vv \| \leq \| \vu \| + \| \vv \| .
$$
Lecture notes (this page) available via link from Course Overview unit on Brightspace. Direct link: https://jdc.math.uwo.ca/l. (That's an "ell" at the end.)
Today we finish Section 1.2. Read Section 1.3 for next class. Work through suggested exercises.
Homework 1 due today at 11:55pm. See course outline for flexibility in homework submission. Homework 2 now available, and due on Sept 20.
Next office hour: Today, 2:30-3:30, MC130 (Middlesex College). Drop by with any questions! The Help Centre will probably start on Monday.
Definition: The dot product or scalar product of vectors $\vu$ and $\vv$ in $\R^n$ is the real number defined by $$ \vu \cdot \vv := u_1 v_1 + \cdots + u_n v_n . $$
This has familiar properties; see Theorem 1.2.
Definition: The length or norm of $\vv$ is the scalar $\|\vv\|$ defined by $$ \|\vv\| := \sqrt{\vv \cdot \vv} = \sqrt{v_1^2 + \cdots + v_n^2} . $$ A vector of length 1 is called a unit vector.
Theorem 1.5: The Triangle Inequality:
For all $\vu$ and $\vv$ in $\R^n$,
$$
\| \vu + \vv \| \leq \| \vu \| + \| \vv \| .
$$
We define the distance between vectors $\vu$ and $\vv$ by the formula
$$
\begin{aligned}
d(\vu, \vv) :=& \| \vu - \vv \| \\[2px] =& \sqrt{(u_1 - v_1)^2 + \cdots + (u_n - v_n)^2}.
\end{aligned}
$$
Theorem 1.4: The Cauchy-Schwarz Inequality: For all $\vu$ and $\vv$ in $\R^n$, $$ | \vu \cdot \vv | \leq \| \vu \| \, \| \vv \| . $$
We can therefore use the dot product to define the angle between two vectors $\vu$ and $\vv$ in $\R^n$ by the formula $$ \cos \theta = \frac{\vu \cdot \vv}{\| \vu \| \, \| \vv \|}, \quad \text{i.e.,} \quad \theta := \arccos \left( \frac{\vu \cdot \vv}{\| \vu \| \, \| \vv \|} \right), $$ where we choose $0 \leq \theta \leq 180^\circ$. This makes sense because the fraction is between -1 and 1.
We have two very different formulas for the dot product: $$ \begin{aligned} \vu \cdot \vv &= u_1 v_1 + \cdots + u_n v_n , \\[3px] \vu \cdot \vv &= \| \vu \| \, \| \vv \| \, \cos \theta . \end{aligned} $$ The second one has important implications. For example, the dot product doesn't change if you rotate both vectors in the same way!
Example: Angle between $\vu = [-1, 1, -1, 1]$ and $\vv = [1, 2, 2, 0]$: $$ \begin{aligned} \cos \theta &= \cyc{ex1-1}{\frac{\vu \cdot \vv}{\| \vu \| \, \| \vv \|}} \\[2px] &= \cyc{ex1-2}{\frac{(-1)(1) + (1)(2) + (-1)(2) + 0}{(2)(3)}} \\[2px] &= \cyc{ex1-3}{\frac{-1}{6}} , \end{aligned} $$ so $\cyc{ex1-4}{\theta = \arccos(-1/6) = 1.738 = 99.1^\circ}$.
Example: If $\vu = [1, 2, 3]$ and $\vv = [1, 1, -1]$ in $\R^3$, then $\vu \cdot \vv = 1 \cdot 1 + 2 \cdot 1 + 3 \cdot (-1) = 1 + 2 - 3 = 0$, so $\vu$ and $\vv$ are orthogonal.
Question: Are $\vu = [-2, 1, 4]$ and $\vv = [3, 2, 0]$ orthogonal?
Note that we do not need to do the full computation of the angle to answer this!
Example: For any $x$ and $y$, $[x,y]$ and $[y,-x]$ are orthogonal since their dot product is $xy + y(-x) = 0$.
Pythagorean theorem in $\R^n$: If $\vu$ and $\vv$ are orthogonal, then $$ \| \vu + \vv \|^2 = \| \vu \|^2 + \| \vv \|^2 . $$
Explain on board, using Theorem 1.2.
This applet is useful for understanding projections as well.
Example: If $\vu = [-1, 1, 0]$ and $\vv = [1,2,3]$ then $$ \kern-4ex \begin{aligned} \proj_\vu(\vv) = \cyc{ex2-1}{\frac{\vu \cdot \vv}{\vu \cdot \vu} \vu} &= \cyc{ex2-2}{\frac{-1+2+0}{1+1+0} [-1, 1, 0]} \\ &= \cyc{ex2-3}{\frac{1}{2} [-1,1,0]} = \cyc{ex2-4}{[-\!\frac{1}{2}, \frac{1}{2}, 0]} \end{aligned} $$
True/false: If $\vu$, $\vv$ and $\vw$ are vectors in $\R^n$ such that $\vu \cdot \vv = \vu \cdot \vw$ and $\vu \neq \vec 0$, then $\vv = \vw$.
True/false: If $\vv$ and $\vw$ are vectors in $\R^n$ and $c$ is a non-zero scalar with $c \vv = c \vw$, then $\vv = \vw$.
True/false: If $\vu$ is orthogonal to both $\vv$ and $\vw$, then $\vu$ is orthogonal to $2 \vv + 3 \vw$.
You only answer true if a statement is always true. You justify this answer by giving a general explanation of why it is always true, not just an example where it happens to be true.
You answer false if a statement can in some case be false. You justify this answer by giving an explicit example where the statement is false.
Question: Suppose I tell you that $\vu \cdot \vv = 1/2$ and $\vu \cdot \vw = -1$. What is $\vu \cdot (2 \vv + 3 \vw)$?
Question: Does $\proj_{\vu}(\vv)$ always point in the same direction as $\vu$?
Example 4-1: (Board). If $\|\vu\| = 2$, $\|\vv\| = 3$ and $\vu \cdot \vv = 4$, find $\|\vu + \vv\|$. Ideas?
Example 4-2: (Board). Find scalars $k$ so that $[k^2, 3]$ and $[k, 1]$ are parallel.
Example 4-3: (Board). Determine last vertex of parallelogram.