Vectors
The Basics
Let’s start this section off with a quick discussion on what vectors are used for.
Vectors are used to represent quantities that have both a magnitude and a direction.
Good examples of quantities that can be represented by vectors are force and
velocity. Both of these have a direction and a magnitude.
Let’s consider force for a second. A force of say 5 Newtons that is applied in a
particular direction can be applied at any point in space. In other words, the point
where we apply the force does not change the force itself. Forces are independent of
the point of application. To define a force all we need to know is the magnitude of the
force and the direction that the force is applied in.
The same idea holds more generally with vectors. Vectors only impart magnitude and
direction. They don’t impart any information about where the quantity is applied.
This is an important idea to always remember in the study of vectors.
In a graphical sense vectors are represented by directed line segments. The length of
the line segment is the magnitude of the vector and the direction of the line segment is
the direction of the vector. However, because vectors don’t impart any information
about where the quantity is applied any directed line segment with the same length
and direction will represent the same vector.
Consider the sketch below.
Each of the directed line segments in the sketch represents the same vector. In each
case the vector starts at a specific point then moves 2 units to the left and 5 units up.
The notation that we’ll use for this vector is,
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View Full Documentand each of the directed line segments in the sketch are called
representations
of the
vector.
Be careful to distinguish vector notation,
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 Fall '08
 prellis
 Linear Algebra, Vectors, Euclidean space, Standard basis

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