Question 1203668
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It appears there might be a typo. 
I think the portion <font color=blue>Find all real numbers c ∈ R</font> should be <font color=blue>Find all real numbers k ∈ R</font>


e1 = (1,0,0)
e2 = (0,1,0)
e3 = (0,0,1)


v1 = -1*e1 + 2*e2 + k*e3
v1 = -1*(1,0,0) + 2*(0,1,0) + k*(0,0,1)
v1 = (-1,0,0) + (0,2,0) + (0,0,k)
v1 = (-1+0+0, 0+2+0, 0+0+k)
v1 = (-1,2,k)


Follow similar steps to find that
v2 = -1*e1 + k*e2 + 2*e3
v2 = (-1,k,2)


We want to have vectors v1 and v2 to be orthogonal. 
In other words, we want the vectors to be perpendicular to each other. 
This occurs if and only if the dot product of said vectors is 0.


v1 dot v2 = (-1,2,k) dot (-1,k,2)
v1 dot v2 = (-1)*(-1) + 2*k + k*2
v1 dot v2 = 1 + 2k + 2k
v1 dot v2 = 1 + 4k
1 + 4k = 0
4k = -1
<font color=red>k = -1/4 is the final answer</font>
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