How Do Dot Products Reflect Vector Projections?

AI Thread Summary
The dot product of two vectors can be calculated using both component multiplication and the cosine of the angle between them. This relationship highlights that the dot product reflects the projection of one vector onto another. The confusion arises from understanding how the scalar projection relates to the direction of the vectors involved. Essentially, the dot product provides a measure of how much one vector extends in the direction of another. Clarifying the concept of projections can enhance understanding of this mathematical relationship.
Pochen Liu
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I know that a dot product of 2, 2 dimension vectors a, b =

(ax * bx) + (ay * by)

but it also is equal to

a*bCos(θ)

because of "projections". That we are multiplying a vector by the 'scalar' property of the other vector which confuses me because that projection is in the direction of the other vector, therefore no scalar, so I cannot see how these two produce the same outcome.

What am I missing intuitively?
 
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