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Unformatted text preview: is a
vector function. ~
Example 2: Fluid velocity is a vector function and one example is V =
^
^ ; y^ + 0k
xi j B. The Dot and Cross Product
There are two types of multiplication of vectors one result is a scalar(i.e.
a number) and the other is a vector. ~~
The Dot Product: A B =j A jj B j cos
The resultant of the dot product is a scalar.
2 y
A = (1,3,2) (1,0,0) x
(0,0,2) z Figure 2: Sketch of a vector in threedimensional space as in example 1. A θ B
Figure 3: Sketch of the geometry associated with the dot product.
3 ~
~
~~
Example 3: Given: A = (1 0 0), B = (1 2 0). Find: A B . Solution: The dot product is obtained by multiplying the magnitude of each
vector together and multplying by the cosine of the angle between them:
~~ p
A B = 1 5 p15 = 1
Alternatively, we could multiply each component of each vector together
and add to get the same answer:
~~
A B = A1B1 + A2B2 + A3B3 ~~ ~~
Note that A B = B A.
~~
The Cross Product: A B The easiest way to calculate the cross product is to view it like a determinant:
^^k
ij^
~ B = A1 A2 A3 = (A2B3;A3B2)^;(A1B3;A3B1)^+(A1B2;A2B1)k
~
^
A
i
j
B1 B2 B3
Note that the result of the cross product is a vector. The magnitude of the
r...
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 Spring '08
 ZOHAR

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