What quadrant is each vector in?
Are any of the vectors equal–if so which?
A scalar is a quantity that has a size, but no direction. A scalar can be positive or negative. Scalar arithmetic is the usual stuff you learned through grade school: addition, subtraction, multiplication, division, and raising to a power. We can also take the absolute magnitude of a scalar. Normal algebraic symbols, such as x, m, t, are used for scalars.
A vector is a quantity in which magnitude and direction are both important. We need ways to draw vectors on a diagram, ways to quantify vectors with numbers, and methods to do arithmetic operations on vectors–these include "addition", "subtraction", and "multiplication", we use the same words as for scalars but define the operations differently.
It is easy to describe vectors geometrically with arrows. The magnitude of the vector is indicated by its length, and the direction is the direction on the paper. The algebraic symbol used to indicate a vector is a letter with an arrow above it. In advanced texts a vector is often indicated by making the symbol boldfaced.
Location of a vector is immaterial–only the length and direction count. We will adopt the concepts of Quadrants from geometry, but what is important is in which quadrant a vector points, not where it is located.
Several vectors are shown below, along with a coordinate system.
What quadrant is each vector in?
Are any of the vectors equal–if so which?
We define addition geometrically as placing vectors one after the other with the head of one attached to the tail of the next. The sum vector extends from the first tail to the last head. The order does not matter.
Positive scalar: The direction is unchanged, the length is multiplied by the scalar.
Negative scalar: The direction is reversed, the length is multiplied by the magnitude ot the scalar.
There are two general methods:
Head-to-tail: Multiply one of the vectors by (-1) and add the results. This is shown in the left two diagrams.
Tail-to-tail: Place the vectors tail to tail, then draw the difference starting from the tip of the one that is subtracted, and going to the tip of the other. The right hand diagrams show addition and subtraction.
Now we want to put numbers to vectors, again I will stick with 2D vectors.
Polar representation: 
An example would be (15 m,30°). The usual convention is used where positive angles are measured counter-clockwise. Then any vector can be broken up into the sum of two vectors, one parallel to the x-axis, one parallel to the y-axis.
The scalar lengths of the two red vectors above are given from trigonometry, in equation 3-5 of text. These are called the components of the vector, and can be positive, negative, or zero.
(Polar to Cartesian
conversion)
The Cartesian representation of a two dimensional vector is then just the two components, (ax, ay)
It is more common in physics and engineering to write the Cartesian form with unit vectors,
Cartesian to Polar Conversion. If we know the components we get the polar form from
and 
.
There is one remaining wrinkle: inverse tangent is a function that returns two distinct answers, the (angle) and the (angle + pi). Calculators return only one of these, it is up to you to determine the proper quadrant and pick the proper angle.
e.g. 
Later in the course we will define the scalar product also called the dot product. This takes two vectors and produces a scalar.
At the end of the course we will define the vector product or cross product, that produces a vector from two other vectors.
Division of two vectors is not defined.