If a, b, c are non-coplanar vectors and is a real number, then the vectors a+2 b+3 c, b+4 …

Question

If $\vec a,\vec b,\vec c$ are non-coplanar vectors and $\lambda$ is a real number, then the vectors $\vec a+2\vec b+3\vec c$, $\lambda\vec b+4\vec c$ and $(2\lambda-1)\vec c$ are non-coplanar for:

Accepted Answer

In the basis $\{\vec a,\vec b,\vec c\}$, the triple product of the three given vectors equals the determinant

$$\begin{vmatrix}1&2&3\0&\lambda&4\0&0&2\lambda-1\end{vmatrix} = \lambda(2\lambda-1)$$

The vectors are non-coplanar iff this is nonzero, i.e. $\lambda\ne0,\dfrac12$ --- all except two values.

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