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What do you suppose happens when the vectors are in opposite directions, such as (1, 0)T and (-1, 0)T ?

Answer:

The magnitude of the dot product is negative.


Range of the Dot Product of Two Unit Vectors

Here is a sampling of bu and the dot product with au  = (1.0, 0)T for various angles.

Angle b Result Picture
000° (1.000, 0.000)T 1.000Vectors separated by 0 degrees
015° (0.966, 0.259)T 0.966Vectors separated by 15 degrees
030° (0.866, 0.500)T 0.866Vectors separated by 30 degrees
045° (0.707, 0.707)T 0.707Vectors separated by 45 degrees
060° (0.500, 0.866)T 0.500Vectors separated by 60 degrees
075° (0.259, 0.966)T 0.259Vectors separated by 75 degrees
090° (0.000, 1.000)T 0.000Vectors separated by 90 degrees
105° (-0.500, 0.866)T -0.259Vectors separated 105 by degrees
120° (-0.500, 0.866)T -0.500Vectors separated by 120 degrees
135° (-0.707, 0.707)T -0.707Vectors separated by 135 degrees
150° (-0.866, 0.500)T -0.866Vectors separated by 150 degrees
165° (-0.966, 0.259)T -0.966Vectors separated by 165 degrees
180° (-1.000, 0.000)T -1.000Vectors separated by 180 degrees

The bu in each case is the unit vector represented by (cos θ, sin θ )T .


QUESTION 6:

What do you imagine is the range of values for the dot product of the two unit vectors, au · bu  ?