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7

Vectors

7.1 Vectors in Two Dimensions (2-D)
A vector in 2-D is defined by its length and the angle it makes with a reference
axis (usually the x-axis). This vector is represented graphically by an arrow.
The tail of the arrow is called the initial point of the vector and the tip of the
arrow is the terminal point. Two vectors are equal when both their length and
angle with a reference axis are equal.

7.1.1 Addition
r r r
The sum of two vectors uv+ = w is a vector constructed graphically as fol-
lows. At the tip of the first vector, draw a vector equal to the second vector,
such that its tail coincides with the tip of the first vector. The resultant vector
has as its tail that of the first vector, and as its tip, the tip of the just-drawn
second vector (the Parallelogram Rule) (see Figure 7.1).
The negative of a vector is that vector whose tip and tail have been
exchanged from those of the vector. This leads to the conclusion that the dif-
ference of two vectors is the other diagonal in the parallelogram (Figure 7.2).

7.1.2 Multiplication of a Vector by a Real Number
r
If we multiply a vector v by a real number k, the result is a vector whose
r
r
length is k times the length of , and whose direction is that of if k is pos-v v
itive, and opposite if k is negative.

7.1.3 Cartesian Representation
It is most convenient for a vector to be described by its projections on the
x-axis and on the y-axis, respectively; these are denoted by (v , v ) or (v , v ).
1
2
y
x
In this representation:
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© 2000 by CRC Press LLC
© 2001 by CRC Press LLC