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Answer:

Since u · v  =  |u||v| cos θ and all vectors in this problem have length 1.0, u · v  =  cos θ.

The dot product of au · bu  is the cosine of the angle between au and bu, which can be read off the diagram as 0.866.

The dot product of au · cu  is the cosine of the angle between au and cu, which can be read off the diagram as 0.500.


Filtering out the effect of Length

Normalized Vectors

bu is closer in orientation to au    so au · bu is the larger.

Remember: cos 30°   =  0.866, sin 30°   =  0.5, cos 60°   =  0.5, and sin 60°   =  0.866.

The dot products are:

So by using vectors of length one, the effect of length is removed and the dot product is larger when a small angle separates the vectors.


QUESTION 5:

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