Science:Math Exam Resources/Courses/MATH152/April 2022/Question A08/Solution 2

Following the second hint, consider the three vectors from $(1,-1,2)$ to the points $(1,1,0),\ (2,-1,1),\ {\text{ and }}(0,1,c)$. That these three vectors are linearly dependent is characterized by the fact that the parallelepiped that they span has no volume or, equivalently that the matrix formed by these three vectors has determinant equal to 0:

$$0=\det \left[{\begin{array}{ccc}0&2&-2\\1&0&-1\\-1&2&c-2\end{array}}\right]=0-2\det \left[{\begin{array}{cc}1&-1\\-1&c-2\end{array}}\right]-2\det \left[{\begin{array}{cc}1&0\\-1&2\end{array}}\right]=-2(c-3)-2(2)=-2c+2.$$

The above equation has unique solution $c=1$.