Objectives: Topics
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Dot product of vectors
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Magnitude of vectors and distances in \(\mathbb{R}^n\)
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Orthogonal vectors and complements
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Angles between vectors
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Section 6.1 Dot Product, Length, and Orthogonality
Handout 6.1 Dot Product, Length, and Orthogonality
Objectives: Goals
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Compute (a) dot products, (b) magnitudes, (c) distances in \(\mathbb{R}^n\text{,}\) and (d) angles between vectors.
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Apply theorems involving orthogonal complements, row/column space, and their relationships to linear systems.
Motivating Question
For a matrix \(A\text{,}\) which vectors are orthogonal to all the rows of \(A\text{?}\) To the columns of \(A\text{?}\)
Definition 6.1. Dot Product.
The dot product of two vectors \(\vec u,\vec v\in\mathbb{R}^n\) is:
\begin{equation*}
\vec u\cdot\vec v = \vec u^T\vec v = \begin{bmatrix} u_1 \amp u_2 \amp \cdots \amp u_n \end{bmatrix}\begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix} = u_1v_1+u_2v_2+\cdots+u_n v_n.
\end{equation*}
Example 6.2.
\begin{equation*}
\vec u=\begin{bmatrix}-1\\3\\k\\2\end{bmatrix},\qquad
\vec v=\begin{bmatrix}4\\2\\1\\-3\end{bmatrix}.
\end{equation*}
Theorem 6.3. Basic Identities of the Dot Product.
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(Symmetry) \(\vec u\cdot\vec w = \vec w\cdot\vec u\text{.}\)
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(Linearity) \((\vec v+\vec w)\cdot\vec u = \vec v\cdot\vec u + \vec w\cdot\vec u\text{.}\)
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(Scalars) \((c\vec u)\cdot\vec w = c(\vec u\cdot\vec w) = \vec u \cdot (c\vec w)\text{.}\)
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(Positivity) \(\vec u\cdot\vec u\ge 0\text{,}\) and it equals 0 if and only if \(\vec u=\vec 0\text{.}\)
Definition 6.4. Length (Magnitude).
The length, or magnitude, of \(\vec u\in\mathbb{R}^n\) is
\begin{equation*}
\|\vec u\|=\sqrt{\vec u\cdot\vec u} = \sqrt{u_1^2+u_2^2+\cdots+u_n^2}.
\end{equation*}
Example 6.5.
Example 6.6.
Let \(\lVert\vec u\rVert=5\text{,}\) \(\lVert\vec v\rVert=\sqrt{3}\text{,}\) and \(\vec u\cdot\vec v=-1\text{.}\) Compute \(\lVert\vec u+\vec v\rVert\text{.}\)
\begin{equation*}
\| c\vec v\| = \lvert c\rvert\|\vec v\|.
\end{equation*}
Definition 6.7. Unit Vector.
A vector of length 1 is called a unit vector.
Examples of unit vectors:
\begin{equation*}
\vec e_1=\begin{bmatrix}1\\0\end{bmatrix},\quad
\vec y=\frac{1}{\sqrt{5}}\begin{bmatrix}1\\2\end{bmatrix},\quad
\vec v=\frac{1}{3}\begin{bmatrix}2\\1\\2\end{bmatrix}.
\end{equation*}
Definition 6.8. Distance.
For \(\vec u,\vec v\in\mathbb{R}^n\text{,}\) the distance between them is
\begin{equation*}
\operatorname{dist}(\vec u,\vec v)=\lVert\vec u-\vec v\rVert.
\end{equation*}
Example 6.9.
Compute the distance between \(\vec u=\begin{bmatrix}7\\1\end{bmatrix}\) and \(\vec v=\begin{bmatrix}3\\2\end{bmatrix}\text{.}\)
Definition 6.10. Orthogonal Vectors.
This is equivalent to
\begin{equation*}
\| \vec u+\vec v\|^2
= \|\vec u\|^2 + \|\vec v\|^2.
\end{equation*}
The zero vector is orthogonal to every vector in \(\mathbb{R}^n\text{.}\)
Example 6.11.
Sketch the set of all vectors orthogonal to \(\vec v=\begin{bmatrix}3\\2\end{bmatrix}\text{.}\)
Definition 6.12. Orthogonal Complement.
If \(W\) is a subspace of \(\mathbb{R}^n\text{,}\) a vector \(\vec z\) is orthogonal to \(W\) if \(\vec z\cdot\vec w=0\) for every \(\vec w\in W\text{.}\) The set of all such vectors is the orthogonal complement:
\begin{equation*}
W^\perp=\{\vec z\in\mathbb{R}^n:\vec z\cdot\vec w=0 \;\text{for all}\; \vec w\in W\}.
\end{equation*}
The set \(W^\perp\) is also a subspace of \(\mathbb{R}^n\text{.}\) We have that
\begin{equation*}
\dim(W) + \dim(W^\perp) = n\text{.}
\end{equation*}
Furthermore, for any subspace \(W\text{,}\) we have that \(\left(W^\perp\right)^\perp = W\text{.}\)
Example 6.13.
Suppose \(A=\begin{bmatrix}1 \amp 3\\ 2 \amp 6\end{bmatrix}\text{.}\) Find \(\Nul(A)^\perp\text{.}\)
Example 6.14.
Let \(L\) be the line spanned by \(\vec v=\begin{bmatrix}1\\-1\\2\end{bmatrix}\text{.}\) Since \(L\) is 1-dimensional, its orthogonal complement \(L^\perp\) is a plane. Find an equation for the plane \(L^\perp\text{.}\)
Definition 6.15. Row Space.
The row space \(\operatorname{Row}(A)\) is the subspace spanned by the rows of a matrix \(A\text{.}\)
Facts:
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\(\dim(\operatorname{Row}(A))=\dim(\operatorname{Col}(A))\text{.}\)
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A basis for \(\operatorname{Row}(A)\) is given by the pivot rows of \(A\text{.}\)
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In general, \(\operatorname{Row}(A)\) and \(\operatorname{Col}(A)\) are unrelated spaces.
Example 6.16.
Describe \(\Nul(A) \subseteq \mathbb{R}^n\) in terms of an orthogonal subspace.
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\(\vec x\in\Nul(A)\) if and only if \(\underline{\hspace{2in}}\text{.}\)
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This means \(\vec x\) is \(\underline{\hspace{2in}}\) to each row of \(A\text{.}\)
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Therefore \(\operatorname{Row}(A)\) is \(\underline{\hspace{3in}}\) of \(\Nul(A)\text{.}\)
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\(\dim(\operatorname{Row}(A))+\dim(\Nul(A))=\underline{\hspace{1in}}\text{.}\)
Theorem 6.17. The Four Subspaces.
For any \(A\in\mathbb{R}^{m\times n}\text{:}\)
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\begin{equation*} \Row(A)^\perp = \Nul(A) \quad \text{and} \quad \Nul(A)^\perp = \Row(A) \end{equation*}
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\begin{equation*} \Col(A)^\perp = \Nul(A^T) \quad \text{and} \quad \Nul(A^T)^\perp = \Col(A) \end{equation*}
This geometric structure is often shown using overlapping regions representing the four fundamental subspaces.
Theorem 6.18. Dot Product and Angle.
The dot product satisfies
\begin{equation*}
\vec a\cdot\vec b=\lVert\vec a\rVert\,\lVert\vec b\rVert\cos\theta.
\end{equation*}
Thus, if \(\vec a\cdot\vec b=0\text{,}\) then either one vector is the zero vector or the vectors are orthogonal.
Example 6.19.
Find the angle between \(\vec v = \begin{bmatrix} 3 \\ 4 \end{bmatrix}\) and \(\vec u = \begin{bmatrix} \dfrac{3 - 4\sqrt{3}}{10} \\ \dfrac{4 + 3\sqrt{3}}{10} \end{bmatrix}\text{.}\) Note that \(\vec u\) is a unit vector.
