Vectors

Length of a Vector

\[\|\vec{V}\| = \sqrt{{\vec{V}.x}^2 + {\vec{V}.y}^2 + {\vec{V}.z}^2}\]

Result is a scalar.

Normalizing

\[\hat{V} = {\vec{V} \over { \| \vec{V} \| }}\]

Result is a vector with \(length = 1\).

Addition and Subtraction

\[\begin{align*} \vec{C} & = \vec{A} + \vec{B} & \vec{C} & = \vec{A} - \vec{B} \\ \vec{C}.x & = \vec{A}.x + \vec{B}.x & \vec{C}.x & = \vec{A}.x - \vec{B}.x \\ \vec{C}.y & = \vec{A}.y + \vec{B}.y & \vec{C}.y & = \vec{A}.y - \vec{B}.y \\ \vec{C}.z & = \vec{A}.z + \vec{B}.z & \vec{C}.z & = \vec{A}.z - \vec{B}.z \end{align*}\]

Result is a vector.

Multiplication and Division

With a scalar:

\[\begin{align*} \vec{B} & = \vec{A} \, k & \vec{B} & = \frac{\vec{A}}{k} \\ \vec{B}.x & = \vec{A}.x \, k & \vec{B}.x & = \frac{\vec{A}.x}{k} \\ \vec{B}.y & = \vec{A}.y \, k & \vec{B}.y & = \frac{\vec{A}.y}{k} \\ \vec{B}.z & = \vec{A}.z \, k & \vec{B}.z & = \frac{\vec{A}.z}{k} \end{align*}\]

With another vector:

\[\begin{align*} \vec{C} & = \vec{A} \, \vec{B} & \vec{C} & = \frac{\vec{A}}{\vec{B}} \\ \vec{C}.x & = \vec{A}.x \, \vec{B}.x & \vec{C}.x & = \frac{\vec{A}.x}{\vec{B}.x} \\ \vec{C}.y & = \vec{A}.y \, \vec{B}.y & \vec{C}.y & = \frac{\vec{A}.y}{\vec{B}.y} \\ \vec{C}.z & = \vec{A}.z \, \vec{B}.z & \vec{C}.z & = \frac{\vec{A}.z}{\vec{B}.z} \end{align*}\]

Result is a vector.

Dot Product

\[\vec{A} \cdot \vec{B} = \vec{A}.x * \vec{B}.x + \vec{A}.y * \vec{B}.y + \vec{A}.z * \vec{B}.z\]

Result is a scalar.

Length can be derived from dot product of vector with itself:

\[\|\vec{V}\| = \sqrt{\vec{V} \cdot \vec{V}}\]

Dot product is a commutative operation:

\[\vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{A}\]

Dot product of two unit vectors is the cosine of the angle between:

\[\begin{split}\hat{A} \cdot \hat{B} & = \cos(\theta) \\ \vec{A} \cdot \hat{B} & = \|\vec{A}\| \cos(\theta) \\ \vec{A} \cdot \vec{B} & = \|\vec{A}\| \, \|\vec{B}\| \cos(\theta) \\ \theta & = \arccos(\frac{\vec{A} \cdot \vec{B}}{\|\vec{A}\| \, \|\vec{B}\|})\end{split}\]

Cross Product

\[\begin{split}\vec{C} & = \vec{A} \times \vec{B} \\ \vec{C}.x & = \vec{A}.y \, \vec{B}.z - \vec{A}.z \, \vec{B}.y \\ \vec{C}.y & = \vec{A}.z \, \vec{B}.x - \vec{A}.x \, \vec{B}.z \\ \vec{C}.z & = \vec{A}.x \, \vec{B}.y - \vec{A}.y \, \vec{B}.x\end{split}\]

Result is a vector.

Cross product is anticommutative:

\[\vec{A} \times \vec{B} \neq \vec{B} \times \vec{A}\]\[\vec{C} = \vec{A} \times \vec{B} \iff \vec{B} \times \vec{A} = -\vec{C}\]

Spherical Coordinates

A left-handed coordinate system (z-up) is used below:

\(\theta\) (polar)

Angle perpendicular to xy plane.

\(\phi\) (azimuth)

Angle that lies on xy plane.

Given a unit vector:

\[\begin{align*} \theta & = \arccos(z) & 0 \le \theta \le \pi \\ \phi & = \arctan\left({y \over x}\right) & 0 \le \phi \le 2\pi \\ \end{align*}\]

Spherical coordinates can be converted to a unit vector using:

\[\begin{split}x & = cos(\phi) \, sin(\theta) \\ y & = sin(\phi) \, sin(\theta) \\ z & = cos(\theta)\end{split}\]