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 result:

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

Solution:

$$\begin{gathered} a=(-5,5,3) \\ b=(-2,-4,-5) \\ b=v+w \\ v\parallel a=>v=ka \\ w\perp a=>w\mathrm{.}a=0 \\ -2=v_1+w_1 \\ -4=v_2+w_2 \\ -5=v_3+w_3 \\ {v}_1=-5\cdot k=2.1186 \\ {v}_2=5\cdot k \\ {v}_3=3\cdot k \\ {w}_1\cdot (-5)+w_2\cdot 5+w_3\cdot 3=0 \\ {v}_1+w_1=-2 \\ {v}_2+w_2=-4 \\ {v}_3+w_3=-5 \\ 5k+v_1=0 \\ 5k-v_2=0 \\ 3k-v_3=0 \\ 5w_1-5w_2-3w_3=0 \\ k={\displaystyle \frac{-25}{59}}\doteq -0.423729 \\ {v}_1={\displaystyle \frac{125}{59}}\doteq 2.118644 \\ {v}_2={\displaystyle \frac{-125}{59}}\doteq -2.118644 \\ {v}_3={\displaystyle \frac{-75}{59}}\doteq -1.271186 \\ {w}_1={\displaystyle \frac{-243}{59}}\doteq -4.118644 \\ {w}_2={\displaystyle \frac{-111}{59}}\doteq -1.881356 \\ {w}_3={\displaystyle \frac{-220}{59}}\doteq -3.728814 \end{gathered}$$

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