Objective

We are given 2 vectors u and v . We want to decompose vector u into
2 orthogonal vectors w₁ and w₂ such that w₁ lies on v and w₁ + w₂ = u.

Here's what we know.
w₁ is a scalar multiple of v so it can be written as kv.
If we can find k, we'll have what we need.
w₁ + w₂ = u, and w₂ is orthogonal to v which means that w₂ $ v = 0.

So, w₁ = kv, and u = w₁ + w₂ = kv + w₂
Now, we express u $ v as (kv + w₂)$ v = kv$ v + w₂ $ v
But, w₂ $ v = 0 since they are orthogonal vectors.
Therefore u $ v = kv $ v + 0, and solving for k, we get

so,

Note: the majority of math text books write this formula as

but since v $ v = || v ||², and the formula is easier to remember when
both numerator and denominator are dot products, I prefer the former. Both are correct.

Note: w₁ is called the projection of u on v, denoted proj _vu.
w₂ is called the component of u orthogonal to v.

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