Euclidean Space

Section 6.1 Euclidean Space

Euclidean \(n\)-space is the vector space \(\mathbb{R}^{n}\) equiped a dot product.

Subsection 6.1.1 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 6.1.1.

Let \(\mathbf{u}=(1,2,0,-3)\) and \(\mathbf{v}=(3,-2,4,2)\text{.}\) Find \(\mathbf{u}\cdot \mathbf{v}\text{.}\)

Checkpoint 6.1.2.

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 6.1.2 Length

Lemma 6.1.3.

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 6.1.4.

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 6.1.3 Angle

Lemma 6.1.5.

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 6.1.6.

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 6.1.7.

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 6.1.8.

\(\mathbf{u}\) 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 6.1.9.

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 6.1.4 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 6.1.10.

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 6.1.5 Cross Product

The cross product only works for the vectors in \(\mathbb{R}^{3}\)

Example 6.1.11.

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 \(\mathbf{v}\text{.}\)

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