Orthogonal Projection Into a Line

Project the first vector orthogonally into the line spanned by the second vector.

• (a)  $\begin{pmatrix}2\\ 1\end{pmatrix}$, $\begin{pmatrix}3\\ -2\end{pmatrix}$     (b)  $\begin{pmatrix}2\\ 1\end{pmatrix}$, $\begin{pmatrix}3\\ 0\end{pmatrix}$     (c)  $\begin{pmatrix}1\\ 1\\ 4\end{pmatrix}$, $\begin{pmatrix}1\\ 2\\ -1\end{pmatrix}$     (d)  $\begin{pmatrix}1\\ 1\\ 4\end{pmatrix}$, $\begin{pmatrix}3\\ 3\\ 12\end{pmatrix}$

Project the vector orthogonally into the line.

• (a)  $\begin{pmatrix}2\\ -1\\ 4\end{pmatrix},~{}\text{$\{\hskip 1.00006ptc\begin{pmatrix}-3\\ 1\\ -3\end{pmatrix}\mid c\in\mathbb{R}\hskip 1.00006pt\}$}$     (b)  $\begin{pmatrix}-1\\ -1\end{pmatrix}$, the line $y=3x$

Although pictures guided our development of Definition 1.1, we are not restricted to spaces that we can draw. In $\mathbb{R}^{4}$ project this vector into this line.

Definition 1.1 uses two vectors $\vec{s}$ and $\vec{v}$. Consider the transformation of $\mathbb{R}^{2}$ resulting from fixing

and projecting $\vec{v}$ into the line that is the span of $\vec{s}$. Apply it to these vectors.

• (a)  $\begin{pmatrix}1\\ 2\end{pmatrix}$     (b)  $\begin{pmatrix}0\\ 4\end{pmatrix}$

Show that in general the projection transformation is this.

Express the action of this transformation with a matrix.

Example 1.4 suggests that projection breaks $\vec{v}$ into two parts, $\text{proj}_{[\vec{s}\,]}({\vec{v}\,})$ and $\vec{v}-\text{proj}_{[\vec{s}\,]}({\vec{v}\,})$, that are non-interacting. Recall that the two are orthogonal. Show that any two nonzero orthogonal vectors make up a linearly independent set.

• What is the orthogonal projection of $\vec{v}$ into a line if $\vec{v}$ is a member of that line?

• Show that if $\vec{v}$ is not a member of the line then the set $\{\hskip 1.00006pt\vec{v},\vec{v}-\text{proj}_{[\vec{s}\,]}({\vec{v}\,})\hskip 1.00006pt\}$ is linearly independent.

Definition 1.1 requires that $\vec{s}\,$ be nonzero. Why? What is the right definition of the orthogonal projection of a vector into the (degenerate) line spanned by the zero vector?

Are all vectors the projection of some other vector into some line?

Show that the projection of $\vec{v}\,$ into the line spanned by $\vec{s}\,$ has length equal to the absolute value of the number $\vec{v}\mathbin{\smash{\bullet}}\vec{s}\,$ divided by the length of the vector $\vec{s}\,$.

Find the formula for the distance from a point to a line.

Find the scalar $c$ such that the point $(cs_{1},cs_{2})$ is a minimum distance from the point $(v_{1},v_{2})$ by using Calculus (i.e., consider the distance function, set the first derivative equal to zero, and solve). Generalize to $\mathbb{R}^{n}$.

Let $\vec{p}$ be the orthogonal projection of $\vec{v}\in\mathbb{R}^{n}$ onto a line $\ell$. Show that $\vec{p}$ is the point in the line closest to $\vec{v}$.

Prove that the orthogonal projection of a vector into a line has length less than or equal to that of the vector.

Show that the definition of orthogonal projection into a line does not depend on the spanning vector: if $\vec{s}$ is a nonzero multiple of $\vec{q}$ then $(\vec{v}\mathbin{\smash{\bullet}}\vec{s}/\vec{s}\mathbin{\smash{\bullet}}\vec{s}\,)\cdot\vec{s}$ equals $(\vec{v}\mathbin{\smash{\bullet}}\vec{q}/\vec{q}\mathbin{\smash{\bullet}}\vec{q}\,)\cdot\vec{q}$.

Consider the function mapping the plane to itself that takes a vector to its projection into the line $y=x$. These two each show that the map is linear, the first one in a way that is coordinate-bound (that is, it fixes a basis and then computes) and the second in a way that is more conceptual.

• Produce a matrix that describes the function’s action.

• Show that we can obtain this map by first rotatin…