A vector is a quantity that has magnitude and direction. For example, a velocity of 70 MPH North is a vector quantity because it has a magnitude of 70MPH and a direction of North. The most common vector notations are a bold-faced lower case letter 
and a letter with an overhead arrow 
. In contrast to a vector, a quantity with only magnitude is called a scalar. A vector is distinctly different than a point; however, students often confuse the two since both can be written as a grouping of numbers. The following shows the definition of a 3D vector 
and a 3D point 
: 
The main distinction between a vector and a point is that two points are equivalent only if they share exactly the same coordinates 
, 
, and 
. Two vectors are equivalent if they have the same magnitude and direction. A vector can be visualized graphically as a directed line segment with a starting point and an end point. If the starting point for a vector is not specified it is generally assumed that the starting point is the origin (
). In Figure 1-8, the vector 
has its tail at the origin (
) and its head at the end point 
. Although vector 
starts at point 
and ends at 
, it is equivalent to vector 
since they both have the same magnitude and direction.
and a letter with an overhead arrow . In contrast to a vector, a quantity with only magnitude is called a scalar. A vector is distinctly different than a point; however, students often confuse the two since both can be written as a grouping of numbers. The following shows the definition of a 3D vector and a 3D point : , , and . Two vectors are equivalent if they have the same magnitude and direction. A vector can be visualized graphically as a directed line segment with a starting point and an end point. If the starting point for a vector is not specified it is generally assumed that the starting point is the origin (). In Figure 1-8, the vector has its tail at the origin () and its head at the end point . Although vector starts at point and ends at , it is equivalent to vector since they both have the same magnitude and direction. Object could not be loaded.
Figure 1-8.
Vectors 
and 
and Since we will often refer to a vector with arbitrary 
dimensions (rather than with a specific number of dimensions like in the 3 dimensional vector above), it is useful now to point out the formal notation for a vector of length 
:
The "dots" are ellipses and represent all the values that are assumed to exist between 
and 
. It is common in mathematics to write a finite series as 
because the first value 
specifies the initial subscript. The second value specifies the direction of the series which in this example is increasing. For example if the second subscript had been 
as in the case of 
the sequence would have been decreasing. The final 
specifies that the sequence terminates at the 
value and thus is a finite sequence.
Finally, the other vector you will encounter throughout this text is the zero vector
: 
The zero vector is written as a bold-faced 
or a zero with an arrow 
. You should be aware when you see it that the zero vector is not the same as a single scalar 
.
dimensions (rather than with a specific number of dimensions like in the 3 dimensional vector above), it is useful now to point out the formal notation for a vector of length : and . It is common in mathematics to write a finite series as because the first value specifies the initial subscript. The second value specifies the direction of the series which in this example is increasing. For example if the second subscript had been as in the case of the sequence would have been decreasing. The final specifies that the sequence terminates at the value and thus is a finite sequence.Finally, the other vector you will encounter throughout this text is the zero vector
: or a zero with an arrow . You should be aware when you see it that the zero vector is not the same as a single scalar .