Parallel and orthogonal

I need math help in this problem: a=(-5, 5 3) b=(-2,-4,-5) (they are vectors)
Decompose the vector b into b=v+w where v is parallel to a and w is orthogonal to a, find v and w

Correct answer:

v₁ = 2.1186
v₂ = -2.1186
v₃ = -1.2712
w₁ = -4.1186
w₂ = -1.8814
w₃ = -3.7288

Step-by-step explanation:

a = (- 5, 5, 3) b = (- 2, - 4, - 5) b = v + w v ∥ a = > v = k a w ⊥ a = > w . a = 0 - 2 = v_{1} + w_{1} - 4 = v_{2} + w_{2} - 5 = v_{3} + w_{3} v_{1} = - 5 \cdot k v_{2} = 5 \cdot k v_{3} = 3 \cdot k w_{1} \cdot (- 5) + w_{2} \cdot 5 + w_{3} \cdot 3 = 0 v_{1} + w_{1} = - 2 v_{2} + w_{2} = - 4 v_{3} + w_{3} = - 5 5 k + v_{1} = 0 5 k - v_{2} = 0 3 k - v_{3} = 0 5 w_{1} - 5 w_{2} - 3 w_{3} = 0 k = \frac{- 2 5}{5 9} ≐ - 0.423729 v_{1} = \frac{1 2 5}{5 9} ≐ 2.118644 v_{2} = \frac{- 1 2 5}{5 9} ≐ - 2.118644 v_{3} = \frac{- 7 5}{5 9} ≐ - 1.271186 w_{1} = \frac{- 2 4 3}{5 9} ≐ - 4.118644 w_{2} = \frac{- 1 1 1}{5 9} ≐ - 1.881356 w_{3} = \frac{- 2 2 0}{5 9} ≐ - 3.728814

v_{2} = (- 2.1186) = - 2.1186

v_{3} = (- 1.2712) = - 1.2712

w_{1} = (- 4.1186) = - 4.1186

w_{2} = (- 1.8814) = - 1.8814

w_{3} = (- 3.7288) = - 3.7288

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