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 5kv2=0 3kv3=0 5w15w23w3=0  Pivot:Row1Row4 5k+v1=0 v2+w2=4 v3+w3=5 v1+w1=2 5kv2=0 3kv3=0 5w15w23w3=0  Row5Row1Row5 5k+v1=0 v2+w2=4 v3+w3=5 v1+w1=2 v1v2=0 3kv3=0 5w15w23w3=0  Row653 Row1Row6 5k+v1=0 v2+w2=4 v3+w3=5 v1+w1=2 v1v2=0 0.6v1v3=0 5w15w23w3=0  Pivot:Row2Row4 5k+v1=0 v1+w1=2 v3+w3=5 v2+w2=4 v1v2=0 0.6v1v3=0 5w15w23w3=0  Row5+Row2Row5 5k+v1=0 v1+w1=2 v3+w3=5 v2+w2=4 v2+w1=2 0.6v1v3=0 5w15w23w3=0  Row60.6 Row2Row6 5k+v1=0 v1+w1=2 v3+w3=5 v2+w2=4 v2+w1=2 v3+0.6w1=1.2 5w15w23w3=0  Pivot:Row3Row4 5k+v1=0 v1+w1=2 v2+w2=4 v3+w3=5 v2+w1=2 v3+0.6w1=1.2 5w15w23w3=0  Row5+Row3Row5 5k+v1=0 v1+w1=2 v2+w2=4 v3+w3=5 w1+w2=6 v3+0.6w1=1.2 5w15w23w3=0  Row6+Row4Row6 5k+v1=0 v1+w1=2 v2+w2=4 v3+w3=5 w1+w2=6 0.6w1+w3=6.2 5w15w23w3=0  Pivot:Row5Row7 5k+v1=0 v1+w1=2 v2+w2=4 v3+w3=5 5w15w23w3=0 0.6w1+w3=6.2 w1+w2=6  Row650.6 Row5Row6 5k+v1=0 v1+w1=2 v2+w2=4 v3+w3=5 5w15w23w3=0 0.6w2+1.36w3=6.2 w1+w2=6  Row751 Row5Row7 5k+v1=0 v1+w1=2 v2+w2=4 v3+w3=5 5w15w23w3=0 0.6w2+1.36w3=6.2 2w2+0.6w3=6  Pivot:Row6Row7 5k+v1=0 v1+w1=2 v2+w2=4 v3+w3=5 5w15w23w3=0 2w2+0.6w3=6 0.6w2+1.36w3=6.2  Row720.6 Row6Row7 5k+v1=0 v1+w1=2 v2+w2=4 v3+w3=5 5w15w23w3=0 2w2+0.6w3=6 1.18w3=4.4  w3=1.184.4=3.72881356 w2=260.6w3=260.6 (3.72881356)=1.88135593 w1=50+5w2+3w3=50+5 (1.88135593)+3 (3.72881356)=4.11864407 v3=5w3=5+3.72881356=1.27118644 v2=4w2=4+1.88135593=2.11864407 v1=2w1=2+4.11864407=2.11864407 k=50v1=502.11864407=0.42372881  k=59250.423729 v1=591252.118644 v2=591252.118644 v3=59751.271186 w1=592434.118644 w2=591111.881356 w3=592203.728814
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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