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?