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Besides the usual addition of vectors and multiplication of vectors by scalars, there are also two types of multiplication of vectors by other vectors. One type, the dot product, is a scalar product; the result of the dot product of two vectors is a scalar. The other type, called the cross product, is a vector product since it yields another vector rather than a scalar. As with the dot product, the cross product of two vectors contains valuable information about the two vectors themselves.
The dot product of two vectors
and
denoted
, is
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NOTE that the result of the dot product is a scalar.
Example 1:
Vectors v and u are given by their components as follows u = ( -2 , 3, 4) and v = ( 4 , 6, 1.
Find the dot product
of the
two vectors.
Solution:
1.
2.
3.
4.
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Example 2:
Find || v || first using the definition
and then
using property 3 above where v = (3 , - 4)
Solution :
1. definition:
2. Property 3:
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