Vector v4

Find the vector v4 perpendicular to vectors
v1 = (1, 1, 1, -1), v2 = (1, 1, -1, 1) and v3 = (0, 0, 1, 1)

Result

v4 = (Correct answer is: (1,-1,0,0)) Wrong answer

Solution:

v4 v1=0 v4 v2=0 v4 v3=0   1x+1y+1z1w=0 1x+1y1z+1w=0 0x+0y+1z+1w=0 x=1  1 x+1 y+1 z1 w=0 1 x+1 y1 z+1 w=0 0 x+0 y+1 z+1 w=0 x=1  wxyz=0 w+x+yz=0 w+z=0 x=1  w=0 x=1 y=1 z=0  v4=(1,1,0,0)v_{4} \cdot \ v_{1}=0 \ \\ v_{4} \cdot \ v_{2}=0 \ \\ v_{4} \cdot \ v_{3}=0 \ \\ \ \\ \ \\ 1x+1y+1z-1w=0 \ \\ 1x+1y-1z+1w=0 \ \\ 0x+0y+1z+1w=0 \ \\ x=1 \ \\ \ \\ 1 \cdot \ x+1 \cdot \ y+1 \cdot \ z-1 \cdot \ w=0 \ \\ 1 \cdot \ x+1 \cdot \ y-1 \cdot \ z+1 \cdot \ w=0 \ \\ 0 \cdot \ x+0 \cdot \ y+1 \cdot \ z+1 \cdot \ w=0 \ \\ x=1 \ \\ \ \\ w-x-y-z=0 \ \\ w+x+y-z=0 \ \\ w+z=0 \ \\ x=1 \ \\ \ \\ w=0 \ \\ x=1 \ \\ y=-1 \ \\ z=0 \ \\ \ \\ v_{4}=(1,-1,0,0)



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