The Basics of Vectors

To begin, let’s define what a vector and scalar is. James Stewart, in his book Calculus: Concepts & Concepts, said that a vector is a quantity that has direction and magnitude. In contrast, a scalar (measurements, numerical values, etc.) only has magnitude. Vectors are commonly written in the notation (a1,a2) for a two dimensional plane. a1 and a2 are known as components of a vector. Vectors can have more than two components if you are working in a more than two-dimensional system—hence, there is no limit to the number of components that you can have (Stewart, 2005, pg. 642-6450). Be sure not to confuse this vector notation with an ordered pair. You can differentiate between an ordered pair and vector by context. The following is an example of a basic vector. Notice how it has a direction of 45 degrees from the x-axis and a magnitude of 2.83 cm. It can also be written in component form as (2,2).



Check out the following Youtube video that is an excert from the movie Despicable Me:

http://www.youtube.com/watch?v=A05n32Bl0aY (DespicableMeMovie, 2010)

To further illustrate the difference between a vector and a scalar, check out the following link to a websight that was created by NASA. As you read it, look for examples of only vector quantities and only scalar quantities. Also, look for what the difference is between the two.

http://www.grc.nasa.gov/WWW/k-12/airplane/vectors.html ("Scalars and vectors," 2010)

Stewart continued to say that a vector is usually named by a bold letter. For example, we could name the above vector u instead of BA (Stewart, 2005, pg 642). He also said that the notation |u| or ||u|| is commonly used to denote the magnitude of a vector. The formula for finding the magnitude of a general vector v is |v|=(a1^2 +a2^2)^1/2. For a vector with more than two components you would just continue adding the square of all of them in the parenthesis (Stewart, 2005, pg.643).

A vector is a is made up of component vectors that are parallel to the x-axis and y-axis. When these vectors are added together, the vector formed is a. For example, in the following illustration, the blue vector is the main vector and the two red vectors are the blue vectors component vectors. When added together, they equal the blue vector.



Vector Addition, Subtraction, and Scalar Multiplication

Just as with numbers and polynomials, we can do a variety of arithmetic operations on vectors. You can multiply a vector by a number which then acts as a scalar to that vector. You can also add and subtract vectors from each other. To do so, you can apply the Triangle Law. The following link to a GeoGebra applet was created by me to demonstrate this law as well as the Parallelogram law which is just an extension of the Triangle law, but demonstrates that addition and subtraction of vectors are commutative. It also demonstrates the concept of scaling a vector. Play with it a while and see if you can discover a way to add and subtract vectors geometrically and algebraically. Also, look for what scaling a vector does to the vector.

Triangle Law, Parallelogram Law, and Scalar Multiplication Background

Stewart gave eight vector properties. They are listed in the following table (note: The bolded letters are vectors and the unbolded are scalars. Also note that the bolded zero, known as the zero vector, is a vector with all the components equal to zero):

1.a+b=b+a	2.a+(b+c)=(a+b)+c
3.a+0=a	4.a+(-a)=0
5.c(a+b)=ca+cb	6.(c+d)a=ca+da
7.(cd)a=c(d,a)	8.1a=a

(Stewart, 2005, pg. 646)

On top of being able to add, subtract, and complete scalar multiplictaion on vectors, there are two other useful operations that can be performed on vectors. They are the Dot Product and the Cross Product.

The Dot Product

Stewart listed the following two formulas for the dot product, along with a unique feature and some special properties that are associated with it. The formula for the dot product in component form is: (a1, a2,...,aN) dot (b1, b2,...,bN) = a1*b1+a2*b2+...+aN*bN.
The formula for the dot product that is not in component form is: a dot b=|a||b|Cos(T) where T is represents and angle.

One unique feature of the dot product to keep in mind is that we can tell that vectors a and b are perpendicular to each other if their cross product is equal to zero.

Some useful properties of the Dot Product are (Note: The bolded letters are vectors and the unbolded letters are scalars.):

1.a dot a=|a|^2

2.a dot b=b dot a

3.a dot (b+c)=a dot b+a dot c

4.(ca) dot b=c(a dot b)=a dot (cb)

5. 0 dot a=0

(Stewart, 2005, pg 651-654)

The Cross Product

Stewart stated that the following formula will calculate the cross product of two vectors with three components. It is:


(Stewart, 2005, pg 661)
He also defined the formula for the cross product that is not in component form, gave two unique characteristics of the cross product, and listed four properties of the cross product. The formula for the cross product that is not in component form is:

aXb=(|a||b|sin(T))n where T is the angle between 0 and pi and n is the unit vector perpendicular to both a and b.

Two unique characteristics of the cross product to keep in mind are that 1. a X b is orthogonal to vectors a and b and 2. Two vectors a and b are parallel if and only if a X b=0.

Some useful properties of the Cross Product are (Note: The bolded letters are vectors and the unbolded letters are scalars.):

1. a X b = -b X a

2. (ca) X b=c(a X b)=a X (cb)

3. a X (b +c)=a X b + a X c

4. (a+b) X c = a X c + b X c,

(Stewart, 2005, pg. 657-659)

The Unit Vector

The unit vector should definitly not be forgotten in our list of basics about vectors as it is referred to much in vector analysis. The unit vector is a vector with the length of 1. All vectors have a unit vector. Any vector that is not a unit vector can be found by multiplying the unit vector by some constant. To find a unit vector of a given vector, Stewart says divide all the components of the said vector by its magnitude (Stewart, 2005, pg.648).