Given a line ‘a’ and a point ‘b’ we want to find a scaled version(λ)’a’that is as close to ‘b’ as possible.
The right answer is where the line of λa to b is at a right angle to line a.
This line is the difference of the line λa and the line that goes to point ‘b’ , line b –>(b-λa).
proj(a)b = λa –> projecting point b onto a subspace of line a equals scaled line a.
The dot product between both vector lines is zero is the mapping over magnitude as followed:
–> Transpose(a)(b-λa) = 0
Transpose(a)b – Transpose(a)(λa)=0
Transpose(a)(λa)=Transpose(a)b .
λ = Transpose(a)b /Transpose(a)(a)
This is also called mapping over magnitude, because Transpose(a)b is a mapping between two vectors (also known as the dot product of two vectors) and Transpose(a)(a) is in fact the compression scaler of the squared length/ magnitude of the line/vector a.
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