More texts, solutions manuals and packages coming soon, possibly today.

Read Section 1.3, the Exploration on cross products and Section 1.4 (just the part on code vectors) for next class.
Work through recommended homework questions.
Scans of **all** of sections 1.1, 1.2 and 1.3 are available from the
course home page.

Tutorials start **this week**, and include a **quiz** covering until
the end of Section 1.2. It does not cover Section 1.3 or the
Exploration after Section 1.2.

The quizzes last 20 minutes, and are at the end of the tutorial, so
you have time for questions at the beginning.

Questions *similar* to homework questions, but may be slightly different.
There are some true/false questions.

You must write in the tutorial you are registered in.

Different sections have different quizzes, but it is still considered
an academic offense to share information about quizzes.

Office hour: today, 12:30-1:30, MC103B.

Also, if you can't make it to my office hours, feel free to attend Hugo Bacard's
office hours.

Help Centers Monday-Friday 2:30-6:30 in MC 106 starting Thursday.

Lecture notes (this page) available from course web page by
clicking on our class times.

The **vector form** of the equation for $\ell$ is:
$$
\vx = \vp + t \vd
$$
where $\vp$ is the position vector of a point on the line,
$\vd$ is a vector parallel to the line, and $t \in \R$.

This is concise and works in $\R^2$ and $\R^3$.

If we expand the vector form into components, we get the
**parametric form** of the equations for $\ell$:
$$
\begin{aligned}
x &= p_1 + t d_1\\
y &= p_2 + t d_2\\
( z &= p_3 + t d_3 \quad \text{if we are in $\R^3$})
\end{aligned}
$$

The **normal form** of the equation for $\ell$ is:
$$
\vn \cdot (\vx - \vp) = 0 \quad\text{or}\quad \vn \cdot \vx = \vn \cdot \vp ,
$$
where $\vn$ is a vector that is *normal = perpendicular* to $\ell$.

If we write this out in components, with $\vn = [a, b]$, we get the
**general form** of the equation for $\ell$:
$$
a x + b y = c,
$$
where $c = \vn \cdot \vp$.
When $b \neq 0$, this can be rewritten as $y = m x + k$, where
$m = -a/b$ and $k=c/b$.

**Note:** All of these simplify when the line goes through the origin,
as then you can take $\vp = \vec 0$.

**Example:** Find all four forms of the equations for the line in $\R^2$
going through $A = [1,1]$ and $B = [3,2]$.

**Note:** None of these equations is *unique*, as $\vp$, $\vd$ and $\vn$
can all change. The general form is closest to being unique: it is unique
up to an overall scale factor.

**Question:** What are the pros and cons of the different ways of
describing a line?

When expanded into components, it gives the

**Note:** You can read off $\vn$ from the general form.
Two planes are parallel if and only if their normal vectors are parallel.

A plane can also be described in **vector form**.
You need to specify a point $\vp$ in the plane as well as
two vectors $\vu$ and $\vv$ which are parallel to the plane but not parallel to each other.
$$
\vx = \vp + s \vu + t \vv
$$
When expanded into components, this gives the **parametric equations** for a plane:
$$
\begin{aligned}
x &= p_1 + s u_1 + t v_1\\
y &= p_2 + s u_2 + t v_2\\
z &= p_3 + s u_3 + t v_3 .
\end{aligned}
$$
Table 1.3 in the text summarizes this nicely (except for the one typo mentioned above).

It may seem like there are lots of different forms, but really there are two: vector and normal, and these can be expanded into components to give the parametric and general forms.

**Example:** Find all four forms of the equations for the plane in $\R^3$
which goes through the point $P = (1, 2, 0)$ and has normal vector
$\vn = [2, 1, -1]$.

If you get parallel vectors $\vu$ and $\vv$, you need to try again.