\[ \renewcommand{\C}{\mathbb{C}} \renewcommand{\R}{\mathbb{R}} \renewcommand{\Q}{\mathbb{Q}} \]

# 1 Equations of Lines

## 1.1 L02

What is the equation of a line in \(\R^{2}\) passing through the point \(\vec{w}\) and oriented in the direction \(\vec{u}\)?

## 1.2 L02

What is the equation of a line in \(\R^{3}\) passing through the point \(\vec{w}\) and oriented in the direction \(\vec{u}\)?

## 1.3 L02

In \(\R^{2}\) what is the equation of a line passing through the point \([1,2]\) whole angle with the \(x\) axis is \(\frac{\pi}{3}\)?

## 1.4 L02

In \(\R^{2}\) what is the equation of a line passing through the point \([1,2]\) whole angle with the \(x\) axis is \(\frac{\pi}{3}\)?

## 1.5 L02

In \(\R^{2}\) and in \(\R^3\) consider the line \(L\) passing through \(\vec{w}\) with direction \(\vec{u}\) and a line \(L'\) passing through \(\vec{w}\) (same as before) but in the direction of \(\vec{v}\).

- Why can it happen that \(\vec{u}\neq \vec{v}\) but \(L=L'\)?
- Give an example of that happening.

## 1.6 L02

In \(\R^{2}\) and in \(\R^3\) consider the line \(L\) passing through \(\vec{w}\) with direction \(\vec{u}\) and a line \(L'\) passing through \(\vec{v}\) (possibly different from \(\vec{w}\)) but in the same direction \(\vec{u}\).

- Why can it happen that \(\vec{w}\neq \vec{v}\) but \(L=L'\)?
- Give an example of that happening.

# 2 Dot product

## 2.1 (conceptual) L02

Reason in \(\R^2\).

Let \(L\) be the line through the origin in the direction of \(\vec{u}\) and suppose that \(\vec{u}\) is normalized. The formula we saw in class for projecting a vector \(\vec{v}\in\R^2\) onto the line \(L\) is

\begin{equation*} \vec{v}\mapsto \big(\vec{v}\cdot \vec{u}\big) \vec{u}. \end{equation*}The vector \(-\vec{u}\) is also a unit vector tnat generates the line \(L\) so the formula above with \(\vec{-u}\) instead of \(\vec{u}\) should give the same exact result.

- Does it?
- Can you justify it with algebra?
- Can you justify it with geometry?
- Does the same hold in \(\R^3\)?
- If \(\theta\) is the angle between \(\vec{u}\) and \(\vec{v}\), what is the angle between \(\vec{-u}\) and \(\vec{v}\)?

## 2.2 (conceptual) L02

We have done in class the formula to project \(\vec{v}\) onto the line \(L\) passing through the origin in the direction of \(\vec{u}\).

Reason on \(\R^2\) for simplicity. Same applies to \(\R^3\)

Reasoning geometrically, what would the question "Find the projection of \(\vec{v}\) onto a line \(L\) not passing through the origin" mean? What are the problems? What are the implicit assumptions?

## 2.3 L02

Fix \(\vec{u}\in\R^2\). For example take \(\vec{u}=[1,2]\).

- Show that the set

is a line passing through the origin.

- Describe the above line it using the parameterization \(L=\{t\vec{w} \colon t\in\R\}\) (find \(\vec{w}\)).
- Show that the set

is a line **NOT** passing through the origin.

- Describe the above line it using the parameterization \(L=\{t\vec{w} +\vec{\alpha}\colon t\in\R\}\) (find \(\vec{w}\) and \(\vec{\alpha}\)).

# 3 Planes in \(\R^3\)

## 3.1 L03

- In \(\R^3\), what is the set \(\{\vec{v}\colon \vec{v}\cdot [1,0,0]=0\}\)?
- In \(\R^3\), what is the set \(\{\vec{v}\colon \vec{v}\cdot [0,1,0]=0\}\)?
- In \(\R^3\), what is the set \(\{\vec{v}\colon \vec{v}\cdot [0,0,1]=0\}\)?

## 3.2 L03

Working in \(\R^3\), fix a vector \(\vec{u}\in\R^3\), for example \(\vec{u}=[1,-2,2]\)

- In \(\R^3\), Is the set \(\{\vec{v}\colon \vec{v}\cdot\vec{u}=0\}\) a plane, a line, the whole space \(\R^3\), neither of these options, or "it depends on \(\vec{u}\)".

## 3.3 L03

- Working in \(\R^3\), why can any plane passing through the origin be described as \(\{\vec{v}\colon \vec{v}\cdot\vec{u}=0\}\) for some \(\vec{u}\in\R^3\).
- What is the meaning of \(\vec{u}\) in the point above, geometrically, in terms of the plane?
**harder**How would you describe a plane not passing through the origin?