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=v1​+w1​ −4=v2​+w2​ −5=v3​+w3​ v1​=−5⋅ k v2​=5⋅ k v3​=3⋅ k w1​⋅ (−5)+w2​⋅ 5+w3​⋅ 3=0  v1​+w1​=−2 v2​+w2​=−4 v3​+w3​=−5 5k+v1​=0 5k−v2​=0 3k−v3​=0 5w1​−5w2​−3w3​=0  Pivot:Row1↔Row4 5k+v1​=0 v2​+w2​=−4 v3​+w3​=−5 v1​+w1​=−2 5k−v2​=0 3k−v3​=0 5w1​−5w2​−3w3​=0  Row5−Row1→Row5 5k+v1​=0 v2​+w2​=−4 v3​+w3​=−5 v1​+w1​=−2 −v1​−v2​=0 3k−v3​=0 5w1​−5w2​−3w3​=0  Row6−53​⋅ Row1→Row6 5k+v1​=0 v2​+w2​=−4 v3​+w3​=−5 v1​+w1​=−2 −v1​−v2​=0 −0.6v1​−v3​=0 5w1​−5w2​−3w3​=0  Pivot:Row2↔Row4 5k+v1​=0 v1​+w1​=−2 v3​+w3​=−5 v2​+w2​=−4 −v1​−v2​=0 −0.6v1​−v3​=0 5w1​−5w2​−3w3​=0  Row5+Row2→Row5 5k+v1​=0 v1​+w1​=−2 v3​+w3​=−5 v2​+w2​=−4 −v2​+w1​=−2 −0.6v1​−v3​=0 5w1​−5w2​−3w3​=0  Row6−−0.6⋅ Row2→Row6 5k+v1​=0 v1​+w1​=−2 v3​+w3​=−5 v2​+w2​=−4 −v2​+w1​=−2 −v3​+0.6w1​=−1.2 5w1​−5w2​−3w3​=0  Pivot:Row3↔Row4 5k+v1​=0 v1​+w1​=−2 v2​+w2​=−4 v3​+w3​=−5 −v2​+w1​=−2 −v3​+0.6w1​=−1.2 5w1​−5w2​−3w3​=0  Row5+Row3→Row5 5k+v1​=0 v1​+w1​=−2 v2​+w2​=−4 v3​+w3​=−5 w1​+w2​=−6 −v3​+0.6w1​=−1.2 5w1​−5w2​−3w3​=0  Row6+Row4→Row6 5k+v1​=0 v1​+w1​=−2 v2​+w2​=−4 v3​+w3​=−5 w1​+w2​=−6 0.6w1​+w3​=−6.2 5w1​−5w2​−3w3​=0  Pivot:Row5↔Row7 5k+v1​=0 v1​+w1​=−2 v2​+w2​=−4 v3​+w3​=−5 5w1​−5w2​−3w3​=0 0.6w1​+w3​=−6.2 w1​+w2​=−6  Row6−50.6​⋅ Row5→Row6 5k+v1​=0 v1​+w1​=−2 v2​+w2​=−4 v3​+w3​=−5 5w1​−5w2​−3w3​=0 0.6w2​+1.36w3​=−6.2 w1​+w2​=−6  Row7−51​⋅ Row5→Row7 5k+v1​=0 v1​+w1​=−2 v2​+w2​=−4 v3​+w3​=−5 5w1​−5w2​−3w3​=0 0.6w2​+1.36w3​=−6.2 2w2​+0.6w3​=−6  Pivot:Row6↔Row7 5k+v1​=0 v1​+w1​=−2 v2​+w2​=−4 v3​+w3​=−5 5w1​−5w2​−3w3​=0 2w2​+0.6w3​=−6 0.6w2​+1.36w3​=−6.2  Row7−20.6​⋅ Row6→Row7 5k+v1​=0 v1​+w1​=−2 v2​+w2​=−4 v3​+w3​=−5 5w1​−5w2​−3w3​=0 2w2​+0.6w3​=−6 1.18w3​=−4.4  w3​=1.18−4.4​=−3.72881356 w2​=2−6−0.6w3​​=2−6−0.6⋅ (−3.72881356)​=−1.88135593 w1​=50+5w2​+3w3​​=50+5⋅ (−1.88135593)+3⋅ (−3.72881356)​=−4.11864407 v3​=−5−w3​=−5+3.72881356=−1.27118644 v2​=−4−w2​=−4+1.88135593=−2.11864407 v1​=−2−w1​=−2+4.11864407=2.11864407 k=50−v1​​=50−2.11864407​=−0.42372881  k=59−25​≐−0.423729 v1​=59125​≐2.118644 v2​=59−125​≐−2.118644 v3​=59−75​≐−1.271186 w1​=59−243​≐−4.118644 w2​=59−111​≐−1.881356 w3​=59−220​≐−3.728814
v2​=(−2.1186)=−2.1186
v3​=(−1.2712)=−1.2712
w1​=(−4.1186)=−4.1186
w2​=(−1.8814)=−1.8814
w3​=(−3.7288)=−3.7288

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