Vectors in \(\mathbb{R}^n\)

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Definition: Vector

A vector is a quantity that has both length (magnitude) and direction.

Remarks:

• initial point; tail
• terminal point; head; tip
• length; magnitude
• zero vector

Definition: Equivalent vectors

Vectors are said to be equivalent vectors if they have the same magnitude and direction.

Definition: Dimension

The dimension of a vector is the number of entries. Each individual entry of a vector is called a component.

Definition: Magnitude

The magnitude of a vector \(\displaystyle \vec{u}=\left(\begin{array}{c}u_1\\\vdots\\u_n\end{array}\right)\in \mathbb{R}^n\) is

$$\|\vec{u}\| = \sqrt{{u_1}^2+{u_2}^2+\cdots+{u_n}^2}$$

Vector operations

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Definition: Vector Operations

\begin{eqnarray*} \left(\begin{array}{c}u_1\\\vdots\\u_n\end{array}\right) + \left(\begin{array}{c}w_1\\\vdots\\w_n\end{array}\right) &=& \left(\begin{array}{c}u_1+w_1\\\vdots\\u_n+w_n\end{array}\right) \\\\ s\left(\begin{array}{c}u_1\\\vdots\\u_n\end{array}\right) &=& \left(\begin{array}{c}su_1\\\vdots\\su_n\end{array}\right) \end{eqnarray*}

\begin{eqnarray*} \vec{u}+\vec{v} &=& \vec{v} + \vec{u} \\(\vec{u}+\vec{v}) + \vec{w} &=& \vec{u} + (\vec{v} + \vec{w}) \\ \vec{0} + \vec{v} &=& \vec{v} \\\alpha (\beta \vec{v}) &=& (\alpha\beta)\vec{v} \\\alpha(\vec{u}+\vec{v}) &=& \alpha\vec{v} +\alpha\vec{u} \\(\alpha +\beta)\vec{v} &=& \alpha \vec{v} +\beta \vec{v} \\\|\alpha \vec{u}\| &=& |\alpha|\|\vec{u}\| \\\|\vec{u}\| &=&0 \textrm{ if and only if }\vec{u} =\vec{0} \end{eqnarray*}

Unit vectors

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Definition: Unit vectors

A unit of a vector is a vector of length one.

$$\|\vec{u}\|=1$$

Theorem: Unit vectors

A unit of a vector in the direction of a non-zero vector \(\vec{w}\) is the vector.

$$\vec{u}=\frac{\vec{w}}{\|\vec{w}\|}$$

Definition: parallel vectors

Two unit vectors \(\vec{u}\) and \(\vec{w}\) are parallel if

$$\vec{u}=\pm\vec{w}$$

Two non-zero vectors \(\vec{a}\) and \(\vec{b}\) are parallel if their corresponind unit vectors are parallel.

$$\vec{u}\|\vec{w}$$

Theorem:

Every unit vector can be represented as

$$\vec{u}=\left(\begin{array}{c} \cos(\theta)\\ \sin(\theta)\end{array}\right)$$

Remarks:

\(\mathbb{R}^2\)

\(\hat{i}=\vec{i}=\vec{e}_1 = \left(\begin{array}{c}1\\0\end{array}\right)\) and \(\hat{j}=\vec{j}=\vec{e}_2 = \left(\begin{array}{c}0\\1\end{array}\right)\)

\(\mathbb{R}^3\)

\(\hat{i}=\vec{i}=\vec{e}_1 = \left(\begin{array}{c}1\\0\\0\end{array}\right)\), \(\hat{j}=\vec{j}=\vec{e}_2 = \left(\begin{array}{c}0\\1\\0\end{array}\right)\) and \(\hat{k}=\vec{k}=\vec{e}_3 = \left(\begin{array}{c}0\\0\\1\end{array}\right)\)

Dot Product

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Definition: dot (scalar, inner) product

\begin{eqnarray*} \vec{u}=\left(\begin{array}{c}u_1\\\vdots\\u_n\end{array}\right) & & \vec{w}=\left(\begin{array}{c}w_1\\\vdots\\w_n\end{array}\right) \\ \\ \vec{u}\cdot\vec{w} &=& \sum_{i=1}^{n} u_i w_i = u_1w_1+u_2w_2+\cdots+u_nw_n \end{eqnarray*}

Theorem:

\begin{eqnarray*} \vec{u}\cdot\vec{v} &=&\vec{v}\cdot\vec{u} \\ \left(\vec{u}+\vec{v}\right)\cdot\vec{w} & = & \vec{u}\cdot\vec{w} + \vec{v}\cdot\vec{w} \\ \vec{u}\left(\vec{v}+\vec{w}\right) &=& \vec{u}\cdot\vec{v}+\vec{u}\cdot\vec{w} \\ \left(s\vec{u}\right)\cdot\vec{w} &=&\vec{u}\cdot\left(s\vec{w}\right) = s\left(\vec{u}\cdot\vec{w}\right) \\ \|\vec{u}\| &=& \sqrt{\vec{u}\cdot\vec{u}} \end{eqnarray*}

Theorem:

Let \(0\le\theta\le\pi\) denote the angle between vectors \(\vec{u}\) and \(\vec{w}\)

\begin{eqnarray*} \vec{u}\cdot\vec{w} &=&\|\vec{u}\|\|\vec{w}\|\cos\theta \end{eqnarray*}

Definition: orthogonality

Two vectors \(\vec{u}\) and \(\vec{w}\) are orthogonal (perpendicular) if

\begin{eqnarray*}\vec{u}\cdot\vec{w} = 0\end{eqnarray*}

Remark: \(\vec{u}\perp\vec{w}\)

Projections and components

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Definition: vector projection

The vector projection of \(\vec{u}\) onto (non-zero) vector \(\vec{w}\) is a new vector \({\mathrm{proj}}_{\vec{w}}\left(\vec{u}\right)\) that is parallel to \(\vec{w}\) and \({\mathrm{proj}}_{\vec{w}}\left(\vec{u}\right) -\vec{u}\) is orthogonal to \(\vec{w}\).

\begin{eqnarray*}{\mathrm{proj}}_{\vec{w}}\left(\vec{u}\right) = \left(\frac{\vec{u}\cdot\vec{w}}{\vec{w}\cdot\vec{w}}\right)\vec{w} &=& \left(\frac{\vec{u}\cdot\vec{w}}{{\|\vec{w}\|}^2}\right)\vec{w} = \left(\frac{\vec{u}\cdot\vec{w}}{{\|\vec{w}\|}}\right)\frac{\vec{w}}{\|\vec{w}\|} \end{eqnarray*}

Definition: scalar projection

The scalar projection of \(\vec{u}\) onto (non-zero) vector \(\vec{w}\) is the length of the vector projection of \(\vec{u}\) onto \(\vec{w}\)

\begin{eqnarray*}{\mathrm{comp}}_{\vec{w}}\left(\vec{u}\right) &=& \left\|{\mathrm{proj}}_{\vec{w}}\left(\vec{u}\right) \right\| = \frac{\vec{u}\cdot\vec{w}}{{\|\vec{w}\|}} \end{eqnarray*}

Definition: Orthogonal decomposition

The orthogonal decomposition of \(\vec{u}\) in terms of (non-zero) vector \(\vec{w}\) is

\begin{eqnarray*}\vec{u} &=& \underbrace{{\mathrm{proj}}_{\vec{w}}\left(\vec{u}\right) }_{\|\vec{w}}+ \underbrace{\left(\vec{u} - {\mathrm{proj}}_{\vec{w}}\left(\vec{u}\right) \right)}_{\perp\vec{w}} \end{eqnarray*}

Cross Product

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Determinants of order two and three

$$\begin{array}{rcl}\left|\begin{array}{cc}a&b\\c&d\end{array}\right| &=& ad-bc = \overbrace{\det\left(\begin{array}{cc}a&b\\c&d\end{array}\right)}^{\textrm{equivalent notation}} \\\\ \left|\begin{array}{ccc}a_1&a_2&a_3\\u_1&u_2&u_3\\w_1&w_2&w_3\end{array}\right| &=&a_1 \left|\begin{array}{cc}u_2&u_3\\w_2&w_3\end{array}\right| -a_2 \left|\begin{array}{cc}u_1&u_3\\w_1&w_3\end{array}\right| +a_3 \left|\begin{array}{cc}u_1&u_2\\w_1&w_2\end{array}\right| \end{array} $$

Cross Product

$$\begin{array}{rcl} \vec{u}\times\vec{w} &=& \left(\begin{array}{c}u_1\\u_2\\u_3\end{array}\right) \times \left(\begin{array}{c}w_1\\w_2\\w_3\end{array}\right) =\left|\begin{array}{ccc}\vec{e}_1&\vec{e}_2&\vec{e}_3\\u_1&u_2&u_3\\w_1&w_2&w_3\end{array}\right| \\\\ &=&\vec{e_1} \left|\begin{array}{cc}u_2&u_3\\w_2&w_3\end{array}\right| -\vec{e_2} \left|\begin{array}{cc}u_1&u_3\\w_1&w_3\end{array}\right| +\vec{e_3} \left|\begin{array}{cc}u_1&u_2\\w_1&w_2\end{array}\right| \end{array} $$

Direction

$$\begin{array}{rcl} \vec{u}\times\vec{w} & = & - \vec{w}\times\vec{u} \\\\ \vec{u}\perp\left(\vec{u}\times\vec{w}\right)&\textrm{or}& \vec{u}\cdot\left(\vec{u}\times\vec{w}\right) = 0 \\\\ \vec{w}\perp\left(\vec{u}\times\vec{w}\right)&\textrm{or}& \vec{w}\cdot\left(\vec{u}\times\vec{w}\right) = 0 \end{array} $$

Magnitude

$$\begin{array}{rcl} {\left\|\vec{u}\times\vec{w}\right\|}^2 & = & {\left\|\vec{u}\right\|}^2 {\left\|\vec{w}\right\|}^2 -{\left(\vec{u}\cdot\vec{w}\right)}^2 \\\\ {\left\|\vec{u}\times\vec{w}\right\|} &=&{\left\|\vec{u}\right\|} {\left\|\vec{w}\right\|}\sin\theta \end{array} $$

Properties

$$\begin{array}{rcl} \vec{u}\times\vec{w} & = & - \vec{w}\times\vec{u} \\\\ \vec{u}\times{\left(\vec{v}+\vec{w}\right)} & = & \vec{u}\times\vec{v}+\vec{u}\times\vec{w} \\\\ {\left(\vec{v}+\vec{w}\right)}\times\vec{u} & = & \vec{v}\times\vec{u}+\vec{w}\times\vec{u} \\\\ \left(a\vec{u}\right)\times\left( b\vec{w}\right) & = & ab \left(\vec{u}\times\vec{w}\right) \\\\ \vec{u}\times\vec{w} = 0 &\iff & \textrm{either } \vec{u}\|\vec{w}\quad\textrm{or } \vec{u} = \vec{0}\quad\textrm{or } \vec{w} = \vec{0} \end{array} $$

Triple Scalar product

$$\begin{array}{rcl} \vec{u}\cdot\left(\vec{v}\times\vec{w}\right) &=& \left|\begin{array}{ccc}u_1&u_2&u_3\\v_1&v_2&v_3\\w_1&w_2&w_3\end{array}\right| \\\\ &=&\vec{w}\cdot\left(\vec{u}\times\vec{v}\right) \\&=&\vec{v}\cdot\left(\vec{w}\times\vec{u}\right) \end{array} $$