Real Vectors

Definition: Real Column Vector

A real column vector is a column vector $\vec{v}\in {\mathbb{R}}^n$ over the real numbers.

Definition: Real Row Vector

A real row vector is a row vector $\vec{v}\in {\mathbb{R}}^{n\times 1}$ over the real numbers.

Dot Product

Definition: Real Dot Product

The dot product of two real column vectors $\vec{u}={\left[\begin{array}{ccc}{u}_1& \cdots & u_n\end{array}\right]}^{\mathsf{T}}\in {\mathbb{R}}^n$ and $\vec{v}={\left[\begin{array}{ccc}{v}_1& \cdots & v_n\end{array}\right]}^{\mathsf{T}}\in {\mathbb{R}}^n$ is defined as:

$$\vec{u}\cdot \vec{v}\stackrel{\text{def}}{=}\sum _{k=1}^n{u}_k{v}_k$$

TIP

The dot product of two real column vectors is equivalent to the matrix product of the transpose of $\vec{u}$ with $\vec{v}$:

$$\vec{u}\cdot \vec{v}={\vec{u}}^{\mathsf{T}}\vec{v}$$

Theorem: Structure of the Real Vector Space

The dot product is an inner product on ${\mathbb{R}}^n$ $({\mathbb{R}}^n,\mathbb{R},+,\cdot )$.

PROOF

TODO

Theorem: Orthonormality of the Real Standard Basis

The standard basis of the real vector space $({\mathbb{R}}^n,\mathbb{R},+,\cdot )$ is orthonormal.

PROOF

TODO

Cross Product

Definition: Real Cross Product

The cross product $\vec{a}\times \vec{b}$ of two real column vectors $\vec{a}={\left[\begin{array}{ccc}{a}_x& a_y& a_z\end{array}\right]}^{\mathsf{T}}$ and $\vec{b}={\left[\begin{array}{ccc}{b}_x& b_y& b_z\end{array}\right]}^{\mathsf{T}}$ in ${\mathbb{R}}^3$ is defined as the real column vectors

$$\vec{a}\times \vec{b}\stackrel{\text{def}}{=}\left[\begin{array}{c}{a}_y{b}_z-a_z{b}_y\\ a_z{b}_x-a_x{b}_z\\ a_x{b}_y-a_y{b}_x\end{array}\right]$$

Theorem: Magnitude of the Cross Product

The Euclidean norm of the cross product $\vec{a}\times \vec{b}$ is defined via the real sine function as

$$||\vec{a}\times \vec{b}||=||\vec{a}||||\vec{b}||\sin \theta ,$$

where $\theta$ is the angle between $\vec{a}$ and $\vec{b}$.

PROOF

TODO

Theorem: Direction of the Cross Product

The cross product $\vec{a}\times \vec{b}$ is orthogonal to both $\vec{a}$ and $\vec{b}$.

PROOF

TODO

Tip: Right-Hand Rule

The direction of the cross product $\vec{a}\times \vec{b}$ can be determined by the right-hand rule - if you point your index finger in the direction of $\vec{a}$ and your middle finger in the direction of $\vec{b}$ and then your thumb will point in the direction of $\vec{a}\times \vec{b}$ if you stick it out.

Theorem: Self-Product of the Cross Product

The cross product of a real column vector $\vec{a}\in {\mathbb{R}}^3$ with itself is $\vec{0}$.

$$\vec{a}\times \vec{a}=\vec{0}$$

PROOF

$$\vec{a}\times \vec{a}=\left[\begin{array}{c}{a}_y{a}_z-a_z{a}_y\\ a_z{a}_x-a_x{a}_z\\ a_x{a}_y-a_y{a}_x\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$$

Theorem: Anticommutativity of the Cross Product

The cross product is anticommutative:

$$\vec{a}\times \vec{b}=-(\vec{b}\times \vec{a})$$

PROOF

TODO

Theorem: Jacobi Property of the Cross Product (Non-Associativity)

The cross product is not associative but satisfies the Jacobi property:

$$\vec{a}\times (\vec{b}\times \vec{c})-\vec{b}\times (\vec{a}\times \vec{c})+\vec{c}\times (\vec{a}\times \vec{b})=\vec{0}$$

PROOF

TODO

Theorem: Distributivity of the Cross Product

The cross product is distributive over addition:

$$\vec{a}\times (\vec{b}+\vec{c})=\vec{a}\times \vec{b}+\vec{a}\times \vec{c}$$ $$(\vec{b}+\vec{c})\times \vec{a}=\vec{b}\times \vec{a}+\vec{c}\times \vec{a}$$

PROOF

TODO

Theorem: Compatibility of the Cross Product

The cross product is compatible with matrix products:

$$(\lambda \vec{a})\times \vec{b}=\vec{a}\times (\lambda \vec{b})=\lambda (\vec{a}\times \vec{b})$$

PROOF

TODO