Vectors
Caution: LEARN HOW TO ADD VECTORS, IT WILL BE
EXTREMELY IMPORTANT IN SUBSEQUENT PHYSICS
MATERIAL.
Definitions
Scalar  a quantity described by just a number
(e.g. temperature, mass, etc.)
Vector  a quantity described by two numbers, known as
the magnitude and direction.
(e.g. velocity, force, electric field, etc.)
Scalars obey ordinary arithmetic, but for vectors we must invent a new
arithmetic.
Vector Arithmetic
A vector may be represented by an arrow that gives direction and magnitude.
Vector addition is defined by the head-to-tail or parallelogram law. This is
also called the geometric method of vector addition. I will explain this in
class. To add vectors graphically accurately requires a ruler and a protractor.
Another way of representing a vector is to give the length of each of its
components but first we must define vector components. Consider a vector
r (an arrow) that makes an angle of 60 degrees with the positive x-axis and
has a length of 4.
The x-component of r is the vector formed by the projection of r onto the xaxis. The y-component is defined in a similar fashion.
Therefore, in this example,
x = 4 cos 60 and
y = 4 sin 60.
If instead you know the values of x and y for a vector you can then
determine the magnitude, |r| (using Pythagorean theorem) and direction,,:
|r| = sqrt(x2 + y2), and
 = tan-1(y/x).
Definition: The negative of a vector is the vector reversed in direction, (-r).
Definition: Vector Subtraction is defined in terms of vector addition, i.e.
r – s  r + (-s)
Now we can define vector addition in terms of the components of the
vectors. This is called the algebraic method of vector addition. There are
just two steps to finding
r+s
step one
step two
Find the x and y components of both r and s.
Add the x- components of r and s, and do the same for the ycomponents.
Unit vectors are dimensionless, have length one, and lie along the
coordinate axes. For example, i points along +x-axis and j along +y.
We write,
r=xi+yj.
Later we will define the operation of multiplying vectors but this is enough
about vectors for now.
Examples [in class]