Discussion
If the lines \(\frac{x - 4}{m} = \frac{y - 3}{2} = \frac{z + 2}{1}\) and \(\frac{x - 3}{1} = \frac{y - 4}{1} = \frac{z + 3}{m}\) are coplanar, then the values of m are
Answer: B
The values are 1, -4.
Two lines are coplanar when the vector joining a point on each and their two direction vectors are coplanar, so their scalar triple product is zero. Points \((4, 3, -2)\) and \((3, 4, -3)\) give the joining vector \((-1, 1, -1)\):
\[ \begin{vmatrix} -1 & 1 & -1 \\ m & 2 & 1 \\ 1 & 1 & m \end{vmatrix} = -m^{2} - 3m + 4 = 0 \]
so \(m^{2} + 3m - 4 = (m + 4)(m - 1) = 0\), giving \(m = 1\) or \(-4\). The other pairs fail: for example \(m = 4\) makes the determinant \(-24\).
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