Introduction to vectors

There are two types of quantities that you should know of: Scalars and Vectors

scalars

A scalar only has magnitude and is specified by a real number. Examples of such scalars are mass, speed, distance, volume and time. They have one value "attaced" to them and thats it.

$\text{ Temperature: } \ 273K$

In the example above, $\text{ Temperature }$ is the quantity, $273$ is the magnitude, and $K$ is the unit.

vectors

A vector has both magnitude and direction. They are specified by a real number and a direction. Examples of vectors are forces, displacement, velocity and acceleration.

$\text{ Force: } \ 10N \ [ \text{ NW } ]$

In the example above, $\text{ Force }$ is the quantity, $10$ is the magnitude, $N$ is the unit and $[ \text{ NW } ]$ is the direction.

notation

Vectors can be written in two ways.

Typically in physics and other subjects, vectors are written as a variable with an arrow above them: $\vec {a}$, $\vec{F}$, $\vec {d}$

They are also written as two points $\overrightarrow {AB}$ where the vector points from points $A$ to $B$.

Formation of vectors using points

You can form a vector that goes from one point to another by subtracting them.

For two points $A$ and $B$, the vector pointing from $A$ to $B$ is determined by:

$\vec {AB} = B - A$

Say, we have two points $A(1, \ 2, \ 3)$ and $B(4, \ 5, \ 6)$. Then $\vec {AB}$ would be:

$\vec {AB} = B - A = (4, \ 5, \ 6) - (1, \ 2, \ 3) = (3, \ 3, \ 3)$

Representation of vectors

Cartesian representation

You may be most familiar with cartesian representation, since it resembles points on a graph.

The cartesian representation of vectors consists of a ordered pair of coordinates as follows:

$\vec {a} = (x, \ y)$

In this representation, you draw an arrow from the origin $(0, \ 0)$ to the coordinates $(x, \ y)$. The arrow's "head" points at the ordered coordinate pair.

Here is a visualization of cartesian vectors:

Polar representation

The polar representation of a vector consists of a magnitude (length) denoted as $n$ as well as an angle that it makes with the positive x-axis denoted as $\theta$.

$\vec {a} = (n, \ \theta)$

The magnitude is always positive, and the angle is positive in the clockwise direction.

Here is what it looks like on a graph:

Converting representations

Both cartesian and polar coordinates can easily be converted from one to the other.

Cartesian to polar

As stated previously, cartesian coordinates are a pair of $x$ and $y$ coordinates. You can actually use trigonometry to calculate the magnitude and angle of the vector!

Think of the vector as a right angle triangle. For the magnitude, you need to get the hypotenus. For the angle, you need to get the angle between the vector's arrow and the positive x-axis.

To calculate the magnitude of the vector, use the Pythagorean Theorem. Note that $\left \| \vec {v} \right \|$ is notation for the magnitude of a vector. The angle of the vector is calculated by using inverse the tan function ($\tan^-$):

$\left \| \vec {v} \right \| = \sqrt { x^2 + y^2 } \newline \theta = \tan^- (\frac {y} {x})$

example

Convert the following vector $\vec {v} = (3, \ 4)$ to polar coordinates:

answer

$\left \| \vec {v} \right \| = \sqrt { x^2 + y^2 } = \sqrt { (3)^2 + (4)^2 } = \sqrt { 9 + 16 } = \sqrt {25} = 5$

$\theta = \tan^- (\frac {y} {x}) = \tan^- (\frac {4} {3}) = 0.927 \ \text{rad} = 53.1^o$

$\therefore \ \vec{v} = (\left \| \vec {v} \right \|, \ \theta) = (5, \ 53.1^o)$

Polar to cartesian

You can also convert polar coordinates to cartesian. Just like before, if you think of the vector as a right angle triangle, you can use trigonometry to determine the $x$ and $y$ coordinates.

To get the $x$ coordinate, you multiply the magnitude of the vector by the cosine of the angle. For the $y$ coordinate, use the sine of the angle:

$\vec {v}_x = \left \| \vec {v} \right \| \cos \theta \newline \vec {v}_y = \left \| \vec {v} \right \| \sin \theta$

example

Find the $x$ and $y$ coordinates of the following vector:

answer

$\vec {v}_x = \left \| \vec {v} \right \| \cos \theta = (5) \cos 53.1 = 3$

$\vec {v}_y = \left \| \vec {v} \right \| \sin \theta = (5) \sin 53.1 = 4$

$\therefore \ \vec {v} = (x, \ y) = (3, \ 4)$

Types of vectors

There are two type vectors that you should know when solving problems involving them.

Algebraic vectors

Algebraic vectors are the vectors that we have seen up to now. They are vectors that have a reference to some coordinate axes/plane.

Here are a couple examples of algebraic vectors:

Geometric vectors

Geometric vectors are vectors without a reference frame. Their magnitude is identified by the length of the vector itself, and their direction is indicated by the orientation of the vector.

Here is a parallelogram formed by four geometric vectors:

Vectors in the 3rd dimension

As of now, we have only dealt with vectors in second dimension, or $\mathbb{R}^2$. Vectors in the 3rd dimension, or $\mathbb{R}^3$ are very similar.

Instead of two coordinates, they have three:

$\vec {v} = (x, \ y, \ z)$

In fact, vectors are not only limited to $\mathbb{R}^2$ or $\mathbb{R}^3$. They be in any dimension $\mathbb{R}^n$ where $n$ is any real number, however, it may be difficult for you to visualize it, since you live only in the 3rd dimension!

The unit vector

The unit vector is vector that points in a certain direction, and has a magnitude equal to $1$.

You can think of the unit vector as a "direction" vector. Whatever constant you multiply to it, its magnitude will be equal to that constant.

For example, the vector $\vec {v} = (\frac {1} {\sqrt{2}}, \ \frac {1} {\sqrt{2}})$ is a unit vector.

Its magnitude is $\left \| \vec {v} \right \| = \sqrt { (\frac {1} {\sqrt{2}})^2 + (\frac {1} {\sqrt{2}})^2 } = \sqrt { \frac {1} {2} + \frac {1} {2} } = \sqrt {1} = 1$

notation

Vectors can be written as a combination of the unity vectors $\hat {i}$, $\hat {j}$ and $\hat {k}$, where $\hat {i} = (1, \ 0 , \ 0)$, $\hat {j} = (0, \ 1 , \ 0)$ and $\hat {k} = (0, \ 0 , \ 1)$.

For example, the vector $\vec {v} = (2, \ -3, \ 4)$ can be written as $\vec {v} = 2 \hat {i} - 3 \hat {j} + 4 \hat {k}$. This is because the sum of the unit vectors times their coefficents equals $\vec {v}$.

Converting vectors to a unit vector

You can convert any vector to a unit vector by dividing it by its magnitude:

$\vec {u} = \frac { \vec {v} } { \left \| \vec {v} \right \| }$

example

Convert $\vec {v} = (2, \ -3, \ 4)$ to a unit vector.

answer

The magnitude of $\vec {v}$ is $\left \| \vec {v} \right \| = \sqrt { (2)^2 + (-3)^2 + (4)^2 } = \sqrt { 4 + 9 + 16 } = \sqrt {29}$

Now plug it into the formula:

$\vec {u} = \frac {\vec{v}} {\left \| \vec {v} \right \|}$

$\vec {u} = \frac {1} { \sqrt {29} } (2, \ -3, \ 4)$

$\vec {u} = (\frac {2} { \sqrt {29} }, \ \frac {-3} { \sqrt {29} }, \ \frac {4} { \sqrt {29} })$

$\therefore$ The unit vector of $\vec {v}$ is $\vec {u} = (\frac {2} { \sqrt {29} }, \ \frac {-3} { \sqrt {29} }, \ \frac {4} { \sqrt {29} })$