Vectors and Parametric Equations
A vector is a quantity that has both magnitude and direction. For example a displacement vector of 30 meters east could be represented in a variety of ways:
The magnitude of the vector might be represented by absolute value signs around the vector symbol, or just the letter without the boldface.
Vector Addition:
The sum of two vectors, A and B , is a vector C , which is obtained by placing the initial point of B on the final point of A , and then drawing a line from the initial point of A to the final point of B , as illustrated in figure. This is also referred to as the "Tip-to-Tail" method. |
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The operation of vector addition as described here can be written as C = A + B
Vector Subtraction:
Vector subtraction is defined in the following way. The difference of two vectors, A - B , is a vector C that is, C = A - B
or C = A + (-B) .Thus vector subtraction can be represented as a vector addition.
The graphical representation is shown in figure.. Inspection of the graphical representation shows that we place the initial point of the vector -B on the final point the vector A , and then draw a line from the initial point of A to the final point of -B to give the difference C .
Example: Vectors u and v are shown at the right. Use these vectors to sketch the following:
a. 2u + 3v b. u – 2v
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a. |
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b. |
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Example:
If u = (1, –3) and v = (2, 5) find: a. u + v b. u – v c. 2u – 3v
Solution: a. u + v = (1, –3) + (2, 5)
= (1 + 2, –3 + 5) = (3, 2)
b. u – v = (1, –3) – (2, 5)
= (1 – 2, –3 – 5) = (–1, –8)
c. 2u – 3v = 2(1, –3) – 3(2, 5)
= (2, –6) – (6, 15) = (–4, –21)
Representation of a Vector as an Ordered Pair
Example: Write the ordered pair that represents the vector from X (–3, 5) and Y (4, –2). Then find the magnitude of  .
Solution: First, represent  as an ordered pair.
 = (4 – (–3), –2 – 5) or (7, –7)
Then, determine the magnitude of  .
 = 
= 
=  or 
 is represented by the ordered pair (7, –7) and has a magnitude of  units.
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Vector Operations |
The following operations are defined for   , and any real number k.
Addition: 
Subtraction: 
Scalar multiplication:  |
Example:
Let  ,  and  . Find each of the following. Solution: a. 
= 
= 
b. 
= 
= 
c. 
 = 7 
= 
= 
d. 
 = 
= 
= 
Representation of a Vector as an Ordered Triple |
Suppose P 1 ( x 1 , y 1 , z 1 ) is the initial point of a vector in space and P 2 ( x 2 , y 2 , z 2 ) is the terminal point. The ordered triple represents  is  . Its magnitude is given by  . |
Example:
Write the ordered triple that represents the vector from X(5, –3, 2) to Y(4, –5, 6). 
= 
= 
Example: Find an ordered triple that represents  if  and  .
  , 
= 
= 
Example:
Find each inner product if  ,  and  . Is any pair of vectors perpendicular? Solution: a. 
= 14 – 14
= 0
 and  are perpendicular
b. 
= 21 + 70
= 91
 and  are not perpendicular.
c. 
= 6 – 5
= 1
 and  are not perpendicular.
The Angle Between Two Vectors |
 , where 0 ° £ q £ 180 °
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Example:
To the nearest degree, find the measure of q , the angle between the vectors (1, 2) and (–3, 1). Solution:  =  = – 0.1414
Therefore, q » 98 °
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is 
. Its magnitude is given by 
. 
and 
are two vectors, 
and 
, the inner product of 
and 
is defined as 
. 
