BIZNIQUE / LINEAR ALGEBRA / 17
See perpendicular vectors through a dot product
See perpendicular vectors through a dot product
Rotate a direction and watch its component along another direction disappear.
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THE IDEA
Read the example.
Read the example.
Two nonzero vectors are orthogonal when their dot product is zero. Projection onto a unit direction v uses the scalar xᵀv. Orthogonal components let us separate contributions without counting the same direction twice.
uᵀv = ‖u‖₂‖v‖₂ cos θ
Use it well
Use it well
Uncorrelated random variables and geometrically orthogonal vectors are related through centred data, but neither automatically means statistical independence.
TRY IT YOURSELF
A quick check.
A quick check.
What is the dot product of (1, 2) and (2, −1)?
Check your reasoning
Check your reasoning
1×2 + 2×(−1) = 0, so the vectors are orthogonal.
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