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Unformatted text preview: Section 13.3 The Dot Product Multiplying Vectors 1 We have shown that vectors can be added, and we have shown they can be multiplied by a scalar. We now consider the problem of multi- plying two vectors together. There are actually two different ways we shall define to multiply vectors together, each of which have their own individual merits and problems. In this section, we consider the easier of the two called the dot product. 1. Basic Definitions The dot product is a way to multiply two vectors together to obtain a scalar (this is important because multiplication between two objects usually results in a similar object - this is not true with the dot prod- uct). There is both a geometric and algebraic definition for the dot product. In order for the geometric definition to make sense, we need to define the angle between two vectors. Definition 1.1. Suppose vectoru and vectorv are two non-zero vectos. Then there is a unique plane which contains them and we define the angle between vectoru and vectorv to be the smallest angle with 0 lessorequalslant lessorequalslant between vectoru and vectorv in this plane. If either vector is the zero vector, then we define the angle between them to be / 2. Definition 1.2. Suppose vectoru = a vector i + b vector j + c vector k and vectorv = d vector i + e vector j + f vector k are two vectors in 3-space. Then we define the dot product vectoru vectorv as follows: ( i ) (Geometric Definition) vectoru vectorv = || vectoru |||| vectorv || cos( ) where is the angle between them. ( ii ) (Algebraic Definition) vectoru vectorv = ad + be + cf . A similar definition for vectors in 2-space holds (we omit the vector k ). The dot product is easy to work with. Example 1.3. Let vectoru = 2 vector i + 3 vector j + vector k and vectorv = vector i 2 vector j vector k . ( i ) Calculate vectoru vectorv . We have vectoru vectorv = 2 6 1 = 5 . ( ii ) Use this to determine the angle between vectorv and vectoru . We know vectoru vectorv = || vectoru |||| vectorv || cos( ) so it follows that 5 = radicalbig (4 + 9 + 1) radicalbig (1 + 4 + 1)cos ( ) = 14 6 cos ( ) . 1 2 Therefore, we have cos ( ) = 5 14 6 . 5455 , or = arccos ( . 5455) = 2 . 148 . We need two definitions because angles between vectors are usually very difficult to determine, so the geometric definition is difficult to use. However, the algebraic definition is simple to calculate, so we can always use the algebraic definition to determine the dot product and then the geometric definition to determine the angle as we did in the last example. The following result summarizes our idea....
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