# 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
\begin{equation*} \{\vec{v}\in\R^{2} \colon \vec{v}\cdot \vec{u}=0 \} \end{equation*}

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
\begin{equation*} \{\vec{v}\in\R^{2} \colon \vec{v}\cdot \vec{u}=1 \} \end{equation*}

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?