Vectors will be used extensively in this course from beginning to end. If you are not thoroughly familiar with vectors and their addition, subtraction, and multiplication now is the time to get help.
See a classmate, a tutor in the Physics walk-in Tutoring Room, or Dr. Becker during his office


Vectors are quantities that have both magnitude and direction. An direction. An example of a vector quantity is velocity. A velocity has both magnitude (speed) and direction, say 60 miles per hour in a direction due west.

(A scalar quantity is different; it has only magnitude - a time interval of 10 seconds, for example. No direction!)

Fig. 1.8 Vector C is the sum of vectors A and B.



Vectors can be added:
= A + B

Vector C is the vector sum of
vectors A and B, as shown.

Fig. 1.9 The vector sum of two parallel and two anti-parallel vectors.




A vector may be composed of its x- and y-components as shown. Be careful, the components of a vector may be positive or negative depending on whether they are in
the direction of positive or negative axes.

Fig. 1.13 The vector A and its rectangular components.



Vectors may be multiplied in two ways:

1) the scalar (dot) product
o B

2) the vector (cross) product
x B



The scalar (or dot) product of two vectors is defined as
A o B =
A B cos f = AxBx + AyBy + AzBz

Note: The dot product of two vectors is a scalar quantity.

Fig. 1.19 The scalar product of two vectors:    ( A o B)






The vector (or cross) product of two vectors is
written as C = A x B
where the direction of the vector product is given by the right-hand rule as shown in the figure.

Fig. 1.22 The direction of the
cross product of two vectors.



The magnitude of the vector product is given by

C = |C| = |A x B| = AB sin f

Fig. 1.23 The magnitude of the cross product.



© 2013 J. F. Becker