\( \newcommand{\R}{\mathbb{R}} \newcommand{\norm}[1]{\left\lVert #1 \right\rVert} \newcommand{\la}{\left\langle} \newcommand{\ra}{\right\rangle} \newcommand{\bbm}{\begin{bmatrix}} \newcommand{\ebm}{\end{bmatrix}} \newcommand{\orr}{\overrightarrow} \newcommand{\i}{\hat{\imath}} \newcommand{\j}{\hat{\jmath}} \newcommand{\k}{\hat{k}} \newcommand{\dotp}{\boldsymbol{\cdot}} \newcommand{\uu}{\vec{u}} \newcommand{\vv}{\vec{v}} \newcommand{\ww}{\vec{w}} \)

Math 1410, Spring 2020

Dot and Cross Products

January 14, 2020

Sean Fitzpatrick
University of Lethbridge

Warm-up

Given vectors \(\vec{v}=\la 2,-1,3\ra\) and \(\vec{w} = \la -4,5,1\ra\text{,}\) find:

  1. \(\displaystyle \vec{v}+\vec{w}\)
  2. \(\displaystyle 3\vec{v}-2\vec{w}\)
  3. \(\displaystyle \norm{\vec{v}}\)

Scalar multiplication

Recall: Given a real number \(c\) and a vector \(\vec{v}=\la a,b\ra\text{,}\)

\begin{equation*} c\vec{v} = c\la a,b\ra = \la ca,cb\ra\text{.} \end{equation*}
The story in \(\R^3\) is similar:
\begin{equation*} c\la x,y,z\ra = \la cx,cy,cz\ra\text{.} \end{equation*}

Some observations:

  1. For any vector \(\vec{v}\text{,}\) \(0\vec{v}=\vec{0}\) and \((-1)\vec{v}=-\vec{v}\text{.}\)

  2. We have \(\vec{v}+\vec{v}=2\vec{v}\text{.}\)

  3. In general, \(a\vec{v}+b\vec{v}=(a+b)\vec{v}\text{.}\)

  4. Also, \(c(\vec{v}+\vec{w})=c\vec{v}+c\vec{w}\text{.}\)

Scalar multiplication, geometrically

  • \(2\vec{v}=\vec{v}+\vec{v}\text{,}\) so \(2\vec{v}\) is in the same direction as \(\vec{v}\text{,}\) but twice as long.

  • In general, we have:

    Theorem:

    For any vector \(\vec{v}\) and scalar (number) \(c\text{,}\)

    \begin{equation*} \norm{c\vec{v}}=\lvert c\rvert \norm{\vec{v}}\text{.} \end{equation*}

Parallel vectors, unit vectors

  • Definition: Parallel vectors.

    We say that two vectors \(\vec{v}\) and \(\vec{w}\) are parallel if \(\vec{w}=c\vec{v}\) for some scalar \(c\text{.}\)

  • Definition: Unit vector.

    A vector \(\vec{u}\) is a unit vector if \(\norm{\vec{u}}=1\text{.}\)

  • Unit vectors are useful when we care about direction, but not magnitude.

  • Given \(\vec{v}=\la 2,3\ra\text{,}\) what is a unit vector in the direction of \(\vec{v}\text{?}\)

Standard unit vectors

In \(\R^2\text{:}\)

\begin{equation*} \i = \la 1, 0\ra, \quad \j = \la 0,1\ra\text{.} \end{equation*}

In \(\R^3\text{:}\)

\begin{equation*} \i = \la 1,0,0\ra, \quad \j = \la 0,1,0\ra, \quad \k = \la 0,0,1\ra\text{.} \end{equation*}

Using the standard unit vectors

Write the vector \(\vec{v} = \la 4,-7,6\ra\) in terms of the vectors \(\i, \j, \k\text{.}\)

Dot Products

The dot product provides the algebra – geometry bridge.

Definition:

Let \(\vec{v}=\la v_1,v_2\ra, \vec{w}=\la w_1,w_2\ra\) be vectors in \(\R^2\text{.}\) The dot product \(\vv\dotp\ww\) is given by

\begin{equation*} \vv\dotp\ww=v_1w_1+v_2w_2\text{.} \end{equation*}

For \(\vec{v}=\la v_1,v_2,v_3\ra, \vec{w}=\la w_1,w_2,w_3\ra\) in \(\R^3\text{,}\) we similarly have

\begin{equation*} \vv\dotp\ww=v_1w_1+v_2w_2+v_3w_3\text{.} \end{equation*}

Examples

For \(\vec{v}=\la 3,-4\ra, \vec{w}=\la 6,2\ra\text{:}\)

  • Compute \(\vv\dotp\ww\)

  • Compute \(\vv\dotp(3\ww)\)

  • Compute \(3(\vv\dotp\ww)\)

For \(\uu = \la 2,-1,3\ra, \vv=\la -3, -1, 4\ra, \ww = \la 0, 1, -5\ra\text{:}\)

  • Compute \(\uu\dotp \vv + \uu\dotp\ww\)

  • Compute \(\uu\dotp (\vv+\ww)\)

Properties of the dot product

Theorem:

Let \(\uu, \vv, \ww\) be vectors, and let \(c\) be a scalar. Then:

  1. \(\vv\dotp\ww = \ww\dotp\vv\)

  2. \(\uu\dotp(\vv+\ww) = \uu\dotp\vv+\uu\dotp \ww\)

  3. \(\uu\dotp (c\vv) = (c\uu)\dotp \vv = c(\uu\dotp\vv)\)

  4. \(\vv\dotp\vv = \norm{\vv}^2\)

  5. \(\vv\dotp\ww = \norm{\vv}\,\norm{\ww}\cos(\theta)\text{,}\) where \(\theta\) is the angle between \(\vv\) and \(\ww\text{.}\)

Orthogonal vectors

The dot product lets us compute angles between vectors. Example:

\begin{equation*} \vv = \la 2, -1\ra, \ww = \la 3, 2\ra\text{.} \end{equation*}

Most useful for us: when \(\theta = \pi/2\text{.}\)

Definition:

We say that two vectors \(\vv, \ww\) are orthogonal if \(\vv\dotp\ww = 0\text{.}\)

Example

Decide if the triangle with vertices \(P=(1,0,2)\text{,}\) \(Q=(3,-1,0)\text{,}\) \(R = (4, 3, -1)\) is a right-angled triangle.

Orthogonal projection

This is probably the most important application of the dot product in Math 1410. To give it proper attention, we'll hold it over to Thursday's class.

Cross products

  • Defined for vectors in \(\R^3\) only.

  • Produces a vector rather than a scalar.

  • Cross product \(\vv\times \ww\) is orthogonal to both \(\vv\) and \(\ww\text{.}\)

  • Definition: if \(\vv = \la v_1,v_2,v_3\ra, \ww = \la w_1,w_2,w_3\ra\text{,}\)

    \begin{equation*} \vv\times\ww = \la v_2w_3-v_3w_2,v_3w_1-v_1w_3,v_1w_2-v_2w_1\ra\text{.} \end{equation*}

Example

Let \(\uu = \la 3, -1, 2\ra, \vv = \la 0, 2, -1\ra, \ww = \la -2, 0, 4\ra\text{.}\) Find:

  • \(\uu\times \vv\)

  • \(\vv\times \ww\)

  • \(\ww\times \vv\)

Areas and angles

If the angle between \(\vv\) and \(\ww\) is \(\theta\text{,}\)

\begin{equation*} \norm{\vv\times\ww} = \norm{\vv}\,\norm{\ww}\sin\theta\text{.} \end{equation*}
The direction of \(\vv\times\ww\) given by “right-hand rule”.

Useful to note: if \(\vv\) and \(\ww\) form 2 of 4 sides of a parallelogram, that parallelogram has area \(A = \norm{\vv}\,\norm{\ww}\sin\theta\text{.}\)

Examples

Let \(P=(0,2,-1), Q = (3, 1, -2), R = (4, -2, 0), S = (7, -3, -1)\text{.}\)

Verify that the quadrilateral with these vertices is a parallelogram, and find its area.

What about the triangle \(\Delta PQR\text{?}\)