Precalculus

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Rules of Thumb

A vector is a straight line with both magnitude (length) and direction.
Vectors do not have a set starting point. They are just lines which exist and can be moved around like a drag-and-drop feature on a computer.
Scalars only affect magnitude, not direction
Vectors are often given for motion and navigation (North, South, East, and West).
Bearing (navigation) - which direction a ship is heading.
North/South of East/West - somewhere in one of the four respective quadrants.
Due Northeast/Northwest/Southeast/Southwest - exactly 45° from the starting point to the respective quadrant.
Matrices follow the same pattern no matter how many terms there are
Matrices are just arrays of vectors


What to Review

Graph skills
Translating geometric figures (especially straight lines) in coordinate planes
General trigonometry


Vector Form

$\vec{v}=\displaystyle\binom{a}{b}$
$a$=Horizontal Distance
$b$=Vertical Distance
Adding Vectors
In vector form: add horizontally across
$\displaystyle\binom{a}{b}+\displaystyle\binom{c}{d}=\displaystyle\binom{a+c}{b+d}$
Graphically: Tip-to-tail method


Vectors

Draw the two vectors, one starting from the origin and the other starting from the end of the first, draw a new vector stretching from origin to the tip of the second vector; that is your added vector.
Component Vectors
Vectors can be expressed as the sum of two other vectors: component vectors e.g. $\displaystyle\binom{0}{b}+\displaystyle\binom{a}{0}=\displaystyle\binom{a}{b}$


Scalars

Scalars are values which one can multiply vectors by in order to increase of decrease their magnitude
In vector form, multiply values inside grouping symbols by the scalar e.g. $2\displaystyle\binom{a}{b}=\displaystyle\binom{2a}{2b}$
Graphically, increase/decrease the vector’s magnitude appropriately
Angles
Find angles between vectors using trigonometry (you will find arctan to be especially useful)


Matrix Properties

Matrices are of the form
$\begin{bmatrix} a&b&c\\ d&e&f\\ g&h&i \end{bmatrix}$ ($3\times3$) or
$\begin{bmatrix} a&b\\ c&d \end{bmatrix}$ ($2\times2$)
Can be as large as you want


The Determinant

$|A|$ means the determinant of matrix $[A]$. Can only be found for square matrices.
Is calculated by $|A| = ad − bc$ for a $2\times2$ matrix and $|A| = a(ei − fh) − b(di − fg) + c(dh − eg)$ for a $3\times3$ matrix


Inverting Matrices

To find an inverse matrix, $[A]^{-1}$, The following must be true: $[A]\cdot [A]^{-1}=[A]^{-1}\cdot[A]=[I]$
$[I]$ is the identity matrix, a matrix with determinant equal to 1:
$\begin{bmatrix} 1&0\\ 0&1 \end{bmatrix}$ for a $2\times2$ matrix and $\begin{bmatrix} 1&0&0\\ 0&1&0\\ 0&0&1 \end{bmatrix}$ for a $3\times3$ matrix


Multiplying Matrices

By a number (scalar): multiply every value in that matrix by that number just like vector
By another matrix: take the dot product (number of rows of one matrix must equal number of columns of other, otherwise multiplication is impossible. Multiplication is not commutative).
$\begin{bmatrix} a&b&c\\ d&e&f\\ \end{bmatrix}\cdot \begin{bmatrix} h&i\\ j&k\\ l&m \end{bmatrix}=\begin{bmatrix} abchjl&abcikm\\ defhjl&defikm \end{bmatrix}$