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))

Solution:

$$\begin{gathered} v_4\cdot v_1=0 \\ {v}_4\cdot v_2=0 \\ {v}_4\cdot v_3=0 \\ 1x+1y+1z-1w=0 \\ 1x+1y-1z+1w=0 \\ 0x+0y+1z+1w=0 \\ x=1 \\ 1\cdot x+1\cdot y+1\cdot z-1\cdot w=0 \\ 1\cdot x+1\cdot y-1\cdot z+1\cdot w=0 \\ 0\cdot x+0\cdot y+1\cdot z+1\cdot w=0 \\ x=1 \\ w-x-y-z=0 \\ w+x+y-z=0 \\ w+z=0 \\ x=1 \\ w=0 \\ y=-1 \\ z=0 \\ {v}_4=(1,-1,0,0) \end{gathered}$$

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