Question

If i+j+k , i-j+k , and 2i+3j+mk are coplanar, then find m.

Answer 1

Answer:

m = 2

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Step-by-step explanation:

One way is this...

These vectors are coplanar (i.e. the three points are coplanar with the origin) if the vectors are linearly dependent.  This means that the determinant of their coefficients is zero.  So...

$\displaystyle\left|\begin{array}{ccc}1&1&1\\1&-1&1\\2&3&m\end{array}\right| = 0 \\ \\ \\\Rightarrow\left|\begin{array}{ccc}1&1&1\\0&-2&0\\0&1&m-2\end{array}\right| = 0 \\ \\ \\\Rightarrow\left|\begin{array}{cc}-2&0\\1&m-2\end{array}\right| = 0 \\ \\\Rightarrow (-2)(m-2) = 0 \\\Rightarrow m = 2$