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Section 5.1 Euclidean Space
Euclidean
\(n\) -space is the vector space
\(\mathbb{R}^{n}\) equiped a dot product.
Subsection 5.1.1 Dot product in \(\mathbb{R}^{n}\)
Dot product in \(\mathbb{R}^{n}\) .
The dot product of \(\mathbf{u}=\left(u_{1}, u_{2}, \ldots, u_{n}\right)\) and \(\mathbf{v}=\left(v_{1}, v_{2}, \ldots, v_{n}\right)\) is the scalar quantity
\begin{equation*}
\mathbf{u} \cdot \mathbf{v}=u_{1} v_{1}+u_{2} v_{2}+\cdots+u_{n} v_{n}
\end{equation*}
Example 5.1.1 .
Let
\(\mathbf{u}=(1,2,0,-3)\) and
\(\mathbf{v}=(3,-2,4,2)\text{.}\) Find
\(\mathbf{u}\cdot \mathbf{v}\text{.}\)
Checkpoint 5.1.2 .
Let \(\mathbf{u}=(1,0,-3)\) and \(\mathbf{v}=(3,2,1)\text{.}\)
Find
\(\mathbf{u}\cdot \mathbf{v}\text{.}\)
Find a nonzero vector
\(w\) such that
\(\mathbf{w}\cdot \mathbf{u}=0=\mathbf{w}\cdot \mathbf{v}\text{.}\)
Subsection 5.1.2 Length
Lemma 5.1.3 .
Suppose that \(\mathbf{u}=\left(u_{1}, u_{2}, \ldots, u_{n}\right)\neq 0.\) Then
\begin{equation*}
\mathbf{u}\cdot \mathbf{u}=u_{1}^{2}+u_{2}^{2}+\cdots+u_{n}^{2} > 0.
\end{equation*}
Definition 5.1.4 .
The length , or norm , of a vector \(\mathbf{v}=\left(v_{1}, v_{2}, \ldots, v_{n}\right)\) in \(\mathbb{R}^{n}\) is
\begin{equation*}
\|\mathbf{v}\|=\sqrt{\mathbf{v} \cdot \mathbf{v}}=\sqrt{v_{1}^{2}+v_{2}^{2}+\cdots+v_{n}^{2}}
\end{equation*}
If
\(\|\mathbf{v}\|=1\text{,}\) then the vector
\(\mathbf{v}\) is called a unit vector.
If \(\mathbf{v}\) is a nonzero vector in \(\mathbb{R}^{n}\text{,}\) then the vector
\begin{equation*}
\mathbf{u}=\frac{\mathbf{v}}{\|\mathbf{v}\|}
\end{equation*}
has length 1 and has the same direction as \(\mathbf{v}\text{.}\) This vector \(\mathbf{u}\) is the unit vector in the direction of \(\mathbf{v}\text{.}\)
Find the unit vector in the opposite direction of
\(\mathbf{v}=(3,-1,2)\text{.}\)
Subsection 5.1.3 Angle
Lemma 5.1.5 .
If \(\mathbf{u}\) and \(\mathbf{v}\) are vectors in \(\mathbb{R}^{n}\text{,}\) then
\begin{equation*}
|\mathbf{u} \cdot \mathbf{v}| \leq\|\mathbf{u}\|\|\mathbf{v}\|
\end{equation*}
where \(|\mathbf{u} \cdot \mathbf{v}|\) denotes the absolute value of \(\mathbf{u} \cdot \mathbf{v}\text{.}\)
Definition 5.1.6 .
The angle \(\theta\) between two nonzero vectors in \(\mathbb{R}^{n}\) is defined as
\begin{equation*}
\theta=\arccos\left(\frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\|\|\mathbf{v}\|}\right)
\end{equation*}
Two vectors
\(\mathbf{u}\) and
\(\mathbf{v}\) in
\(\mathbb{R}^{n}\) are called
orthogonal if
\(\mathbf{u} \cdot \mathbf{v}=0.\)
Theorem 5.1.7 .
If \(\mathbf{u}\) and \(\mathbf {v}\) are vectors in \(\mathbb{R}^{n}\text{,}\) then
\begin{equation*}
\|\mathbf{u}+\mathbf{v}\| \leq\|\mathbf{u}\|+\|\mathbf{v}\|.
\end{equation*}
Theorem 5.1.8 .
\(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal if and only if
\begin{equation*}
\|\mathbf{u}+\mathbf{v}\|^{2}=\|\mathbf{u}\|^{2}+\|\mathbf{v}\|^{2}
\end{equation*}
Example 5.1.9 .
Let
\(\mathbf{u}=(1,2,0,-3)\) and
\(\mathbf{v}=(3,-2,4,2)\text{.}\) Find the angle between
\(\mathbf{u}\) and
\(\mathbf{v}\text{.}\)
Subsection 5.1.4 Distance
The distance between two vectors \(\mathbf{u}\) and \(\mathbf{v}\) in \(\mathbb{R}^{n}\) is
\begin{equation*}
d(\mathbf{u}, \mathbf{v})=\|\mathbf{u}-\mathbf{v}\| .
\end{equation*}
Example 5.1.10 .
Let
\(\mathbf{u}=(1,2,0,-3)\) and
\(\mathbf{v}=(3,-2,4,2)\text{.}\) Find the distance between
\(\mathbf{u}\) and
\(\mathbf{v}\text{.}\)
Subsection 5.1.5 Cross Product
The cross product only works for the vectors in
\(\mathbb{R}^{3}\)
Example 5.1.11 .
Let
\(\mathbf{u}=(1,0,-3)\) and
\(\mathbf{v}=(3,2,1)\text{.}\) Find the
\(\mathbf{u}\times \mathbf{v}\text{,}\) which is a vector orthogonal to both
\(\mathbf{u}\) and
\(\mathbf{v}\text{.}\)
