Standard Vectors

Standard Vectors

Prerequisites Required- Trigonometric Functions

Throughout physics and engineering, few mathematical concepts are as pervasive as vectors. A vector is a mathematical representation for something with both direction and magnitude. One example of a vector is a point to point line, shown in the diagram below; it has both a length and a direction.

Figure 1: Example of a vector

Vectors are not strictly limited to representing lines; they can be used to describe any quantity with direction and magnitude (distance, velocity, acceleration, force, etc.).  

Using the trigonometric functions, any vector can be broken up into components. If you are not familiar with them, I recommend reading the below article.

Trigonometric Functions
Trigonometric functions have critical applications in geometry, physics, and engineering.

Suppose the vector in figure 1 is called $\textbf{L}$. We can determine the components of $\textbf{L}$ in the $x$ and $y$ directions. The $x$ component is the adjacent side and the $y$ component is the opposite side:

$L_x = 1.25\text{cos}(36.87^\circ) = 1$

$L_y = 1.25\text{sin}(36.87^\circ) = 0.75$

The vector can now be drawn showing its components.

Figure 3: Vector example broken into $x$ and $y$ components

This process of splitting a vector into its $x$ and $y$ component is so important that another example is warranted.

Figure 4: Second vector example with $-x$ and $+y$ components

Let's call this second vector $\textbf{L}_2$ and split it into its $x$ and $y$ components:

$L_{2x} = 1.5\text{cos}(148^\circ) = -1.272$

$L_{2y} = 1.5\text{sin}(148^\circ) = 0.795$

Figure 5: Second vector example broken into its $-x$ and $+y$ components

Three Dimensional Vectors

Vectors are not limited to two dimensions. Three dimensional vectors are perfectly valid as well.

Figure 6: Three dimensional vector example broken into $x$, $y$, and $z$ components

The components from the vector in the above diagram are:

$L_x = 3,\:L_y = 1,\:L_z = 2$

$\alpha$, $\beta$, $\gamma$ Representation

One method of describing a 3D vector is by the angles between the vector and the $x$, $y$, and $z$ axes. This is similar to how 2D vectors were first presented in figure 1.

Converting a vector in $\alpha$, $\beta$, $\gamma$ representation to component representation is simple; generate a triangle between each axis and the vector.

Figure 8: $\alpha$, $\beta$, $\gamma$ representation for 3D vector with component triangles

Each component of the vector is the adjacent side of each triangle and can be determined using the cosine function:

$L_x = 3.742\text{cos}(36.699^\circ) = 3$

$L_y = 3.742\text{cos}(57.688^\circ) = 2$

$L_z = 3.742\text{cos}(74.499^\circ) = 1$

$\alpha$, $\beta$ Representation

$\alpha$, $\beta$ representation (not to be confused with $\alpha$, $\beta$, $\gamma$ representation) describes a 3D vector with two angles.

Figure 9: $\alpha$, $\beta$ representation for 3D vector

Converting this representation to components is a little more lengthy but still straightforward:

$L_z = d = 3.754\text{sin}(32.312^\circ) = 2$

$c = 3.754 \text{cos}(32.312^\circ) = 3.1726927$

$L_x = a = 3.1726927\text{cos}(18.435^\circ) = 3$

$L_y = b = 3.1726927\text{sin}(18.435^\circ) = 1$

Representing Vectors

It clearly doesn't make sense to represent every vector with a picture and writing multiple equations to describe a vector is a lot of effort. One reasonable method is to represent a vector as a sum of its components. Consider our first vector example, shown below:

Figure 10: Vector example broken into $x$ and $y$ components

We can mathematically represent this vector as:

$1 \hat{x} + 0.75 \hat{y}$

Note the use of $x$ and $y$ with little hats on them, $\hat{x}$,$\hat{y}$. These symbols are appropriately called "x-hat" and "y-hat", and they represent the $x$ and $y$ directions. The purpose of using these hatted symbols is because the $x$ and $y$ symbols are often used to represent variables; we want to make sure we don't confuse directions for variables.

We can also represent three dimensional vectors.

Figure 11: Three dimensional vector example broken into $x$, $y$, and $z$ components

The above three dimensional vector is represented as:

$3 \hat{x} + 1 \hat{y} + 2 \hat{z}$

A commonly taught alternative to the $\hat{x} $, $\hat{y}$, $\hat{z}$ symbols are $\hat{i} $, $\hat{j}$, and $\hat{k}$ respectively. We might write the above three dimensional vector as:

$3 \hat{i} + 1 \hat{j} + 2 \hat{k}$

I strongly dislike the $\hat{i}$, $\hat{j}$, $\hat{k}$ nomenclature since it conflicts with other symbols we will use in the future. I've shown it here only because it is so often taught.

When referring to something that is a vector, the two most common methods are to make its symbol bold, $\textbf{L}$ or put an arrow above it, $\overrightarrow{L}$. These are both equivalent.

$\textbf{L} = \overrightarrow{L} = 3 \hat{x} + 1 \hat{y} + 2 \hat{z}$

This method of writing a vector (with each component corresponding to $\hat{x}$, $\hat{y}$, $\hat{z}$) is called the component vector form.

The component vector form has the unfortunate side effect of having to draw a lot of little hats.  A much more compact version is the matrix vector form:

$$\overrightarrow{L} = \begin{bmatrix} 3 \\ 1 \\ 2 \end{bmatrix}$$

The first row corresponds to the $\hat{x}$ component, the second to the $\hat{y}$ component, and the third to the $\hat{z}$ component. This is the modern standard vector. A list of numbers set inside brackets.

Vector Positions

While a vector contains magnitude and direction information, it does not contain position information. The below vectors $\overrightarrow{L_1}$ and $\overrightarrow{L_2}$ are equivalent, both equal to $\begin{bmatrix} 1 \\ 1.5 \end{bmatrix}$.

Figure 12: Demonstration of vector position independence

To describe the position of a vector requires another vector. The start of vector $\overrightarrow{L_1}$ is located at $\begin{bmatrix} 0 \\ 0 \end{bmatrix}$ while the start of vector $\overrightarrow{L_2}$ is located at $\begin{bmatrix} -0.5 \\ 0.5 \end{bmatrix}$.

Adding and Subtracting Vectors

To add vectors, we put them end to end and add their components.

Figure 13: Graphical demonstration of vector addition

$$\overrightarrow{L_1} = \begin{bmatrix} 1.4 \\ 0.75 \end{bmatrix},\: \overrightarrow{L_2} = \begin{bmatrix} -0.5 \\ 0.4 \end{bmatrix}$$

$$\overrightarrow{L_1}  + \overrightarrow{L_2 } =   \begin{bmatrix} 1.4 \\ 0.75 \end{bmatrix} + \begin{bmatrix} -0.5 \\ 0.4 \end{bmatrix} = \begin{bmatrix} 1.4-0.5 \\ 0.75+0.4 \end{bmatrix} = \begin{bmatrix} 0.9 \\ 1.15 \end{bmatrix}$$

To subtract vectors is similar to adding them... except we subtract their components instead of adding them.

$$\overrightarrow{L_1}  - \overrightarrow{L_2 } =   \begin{bmatrix} 1.4 \\ 0.75 \end{bmatrix} - \begin{bmatrix} -0.5 \\ 0.4 \end{bmatrix} = \begin{bmatrix} 1.4-(-0.5) \\ 0.75-0.4 \end{bmatrix} = \begin{bmatrix} 1.9 \\ 0.35 \end{bmatrix}$$

Lengths and Magnitudes of Vectors

Recall the pythagorean theorem derived in the trigonometric functions article.

Trigonometric Functions
Trigonometric functions have critical applications in geometry, physics, and engineering.

$a^2 + b^2 = c^2$

For a two dimensional vector, this equation remains valid. It's magnitude can be calculated by simple substitution.

Figure 14: Demonstration of 2D vector magnitude calculation

$c^2 = 1.0^2 + 0.75^2 = 1.5626$

$c = 1.25$

When a vector has units of distance (millimeters, inches, etc.), this quantity is called length. When a vector has other units (speed, acceleration, force, etc.), this quantity is called magnitude.

The symbol for length/magnitude is a set of double vertical bars surrounding the vector. For the above vector $\overrightarrow{L}$, it's length is given by:

$||\overrightarrow{L}|| = 1.25$

The length of a 3D vector is easy to derive.

Figure 15: Demonstration of 3D vector magnitude calculation

In the diagram above, 3D vector $\textbf{e}$ can be projected onto the $xy$ plane; we can call this projection (which is also a vector) $\textbf{c}$. Note that $a$, $b$, and $d$ are the $x$, $y$, and $z$ components of $\textbf{e}$.

$||\textbf{c}|| = \sqrt{a^2 + b^2} = \sqrt{3^2 + 1^2} = 3.162$

$||\textbf{e}|| = \sqrt{(||\textbf{c}||)^2 + d^2} = \sqrt{3.162^2 + 2^2} = 3.742$

Through symbolic substitution, we can recognize that:

$$(||\textbf{c}||)^2 = a^2 + b^2$$

$$\downarrow$$

$$(||\textbf{e}||)^2 = (||\textbf{c}||)^2 + d^2$$

$$\downarrow$$

$$\boxed{(||\textbf{e}||)^2 = a^2 + b^2 + d^2}$$

This is is the 3D pythagorean theorem; the square of a vector's magnitude is equal to the sum of its components squared. We can apply this formula to obtain the same answer we did earlier though stacked pythagorean theorems:

$$||\textbf{e}|| = \sqrt{a^2 + b^2 + d^2} = \sqrt{3^2 + 1^2 + 2^2} = 3.742$$

Scalars

We've described a vector as a quantity with a direction and magnitude. A scalar is a quantity that is only a magnitude without direction. Every number ($4$, $-73$, $9$, etc.) is a scalar. While vectors are symbolized with a bold symbol $\textbf{L}$ or symbol with an arrow $\overrightarrow{L}$, scalars are represented by regular symbols ($x$, $y$, $\theta$, $\alpha$, etc.).

Scalar Multiplication

When vectors are multiplied by a scalar, they grow in size accordingly. The below diagram shows a vector $\textbf{b}$ which is equal to $2\textbf{a}$.

Figure 16: Demonstration of vector scalar multiplication

Note that $\textbf{a} = 0.75 \hat{x} + 0.6 \hat{y}$

Multiplying this vector by a scalar is simple:

$2\textbf{a} = 2(0.75 \hat{x} + 0.6 \hat{y}) = 1.5 \hat{x} + 1.2 \hat{y}$

The matrix representation of this is:

$2\textbf{a} = 2\begin{bmatrix} 0.75 \\ 0.6  \end{bmatrix} = \begin{bmatrix} 1.5 \\ 1.2  \end{bmatrix}$

Vector Multiplication

Given the simplicity of scalar multiplication, we might be inclined to attempt multiplication of vectors. Suppose we have two generic vectors:

$\textbf{L}_1 = a \hat{x} + b \hat{y}$

$\textbf{L}_2 = c \hat{x} + d \hat{y}$

Attempting to multiply them together would yield:

$(\textbf{L}_1)(\textbf{L}_2) = (a \hat{x} + b \hat{y})(c \hat{x} + d \hat{y})$

$(\textbf{L}_1)(\textbf{L}_2) = (ac) \hat{x}^2 + (ad)\hat{x}\hat{y} + (bc)\hat{y}\hat{x} + (bd)\hat{y}^2$

What are $\hat{x}^2$, $\hat{x}\hat{y}$, $\hat{y}\hat{x}$, and $\hat{y}^2$!? These are topics for another article, but they are very powerful.