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Table of Contents
Vectors
Vectors are line segments that represent the magnitude and direction of a quantity, which is useful for real life quantities like force and velocity. It is usually described by two points/ordered pairs. Vectors are written with an arrow on top of the two point letters instead of a line like for line segments.
Component Form
The component form of a vector is the individual changes in horizontal and vertical units that make up the combined vector's magnitude and direction. If the vector is graphed, just count the boxes™, otherwise do $y_2 - y_1$ and $x_2 - x_1$ with the ordered pairs they give you for each component. Component form is written with angled brackets (e.g. <2,-1>) to distinguish it from standard ordered pairs.
Magnitude
Magnitude is the length of the vector. It is an absolute value and can be calculated with Pythagorean Theorem1) with the component form values of the vector. Think of the vector as the hypotenuse of a right angle triangle with each side being the horizontal and vertical component form values respectively. Magnitude is written with two pipes beside the vector variable name2).
Direction
The direction of the vector is pretty self-explanatory. It can be found by doing some basic trig on the vector, with theta being the angle at the origin point and between the hypotenuse and adjacent side. Again, think of the vector as a right angle triangle3).
Math with Vectors
Math with vectors is super simple! To add two vectors together, just add their component values together4). For multiplication with a scalar value, just multiply in the value to both horizontal and vertical components 5). To subtract, scalar multiply the second value by -1 and then add.
Dot Product
The dot product of two vectors is just multiplying the two together. The answer is a scalar value. Take the vectors <x, y> and <j, k>. The dot product of the two would be $xj + yk$. Dot product is useful to find the angle between two vectors.
Angle Between Two Vectors
Take the $\arccos$ of the dot product of the two vectors divided by the magnitudes of the vectors multiplied together. In other words, if vectors $v$ and $w$ exist, the angle $\theta$ between them would be $\theta = \arccos {\frac{v \cdot w}{||v||||w||}}$. The alternative equation for this would be $v \cdot w = ||v||||w||\cos \theta$.
Special Vectors
Zero Vector
The zero vector has its origin at (0,0) and end at (0,0). Its variable name is a bold zero6) and its component form would also be <0,0>.
Unit Vector
A unit vector is a vector with magnitude 1. To normalize a vector to a unit vector, scalar divide7) the vector by its magnitude. Unit vectors can also be made as <$\cos \theta$, $\sin \theta$> where $\theta$ is the angle of the vector. If you multiply unit vectors by the magnitude, you get the vector itself8).
Standard Unit Vectors
There are two:
- i = <1,0>
- j = <0,1>
As these are the the base horizontal and vertical component forms, they can be added together to make any integer vector. For example, <3,2> can be written as 3i + 2j. Sometimes vectors are given to you in terms of i and j. To add these types of vectors together, just substitute them with i and j and simplify via algebra.
Parallel or Orthogonal
Parallel vectors are vector with the same direction but varying magnitude. Check if dividing the two vector's component values give the same ratio for each to see if it is parallel or not. Orthogonal vectors are vectors with a 90 degree angle between them. To check if two vectors are orthogonal, check if their dot product equals zero, if so it is orthogonal.
