The Dot Product: Examples

1. Find the Dot Product of the two vectors: <4,2,5> $\cdot $ <0,2,5>.


Answer:


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2. Using the Dot Product, find the angle between the two vectors: <1,1,1>, <-4,2,-7>.

 

Answer:
 

First, find the magnitude of each vector
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Then find the dot product of the vectors:
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Then following the formula for the angle:
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3. Find the vector projection of <1,5,-3> onto <3,-2,5>.
 


Answer:

 

First, find the magnitude of the second vector:
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Then find the dot product:
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Then put it in the formula:
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