Vector perpendicular

Find the vector a = (2, y, z) so that a⊥ b and a ⊥ c where b = (-1, 4, 2) and c = (3, -3, -1)

Correct result:

y =  5
z =  0

Solution:

a=(2,y,z) b=(−1,4,2) c=(3,−3,−1)  a.b=0 a.c=0    2⋅ (−1)+y⋅ 4+z⋅ 2=0 2⋅ 3+y⋅ (−3)+z⋅ (−1)=0  4y+2z=2 3y+z=6  y=5 z=−9  y=5a=(2, y, z) \ \\ b=(-1, 4, 2) \ \\ c=(3, -3, -1) \ \\ \ \\ a.b=0 \ \\ a.c=0 \ \\ \ \\ \ \\ \ \\ 2 \cdot \ (-1)+y \cdot \ 4+z \cdot \ 2=0 \ \\ 2 \cdot \ 3+y \cdot \ (-3)+z \cdot \ (-1)=0 \ \\ \ \\ 4y+2z=2 \ \\ 3y+z=6 \ \\ \ \\ y=5 \ \\ z=-9 \ \\ \ \\ y=5
z=0

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