Check whether the three vectors $A(2,-1,2),B(1,2,-3),C(3,-4,7) $ are in the same plane

Question

I want to check if three vectors are in the same plane, the vectors being $$A(2,-1,2),B(1,2,-3),C(3,-4,7). $$

What I did so far is to create vector $AB ( -1,3,-5)$ and build the plane equation with the point $A$ $$-1(x-2)+3(y+1)-5(z-2)=0$$ and inserted the point $C$ to check if the equation exists. Is this the right way to do that or I did something wrong?

Thanks!

Answer 1

It would be easier to compute a scalar triple product.. Make three vectors $\hat{u}, \hat{v},\hat{w}$.
The three vectors lay in the same plane iff $$ \hat{u} \cdot (\hat{v} \times\hat{w} ) = 0$$

Answer 2

If $\alpha, \beta,\gamma\in \mathbb{R}^3,$ then we have: $\alpha,\beta,\gamma$ are linearly independent $\iff$ the determinant of matrix $(\alpha,\beta,\gamma)$ is not zero, where $\alpha,\beta,\gamma$ are regarded as column vectors. And we also know that that $\alpha,\beta,\gamma $ is linearly dependent is equivalent to that $\alpha,\beta,\gamma$ are in the same plane. Then it is easy to check $$\begin{vmatrix}2 &-1&2\\1 &2 &-3\\ 3 &-4 &7 \end{vmatrix}=0,$$ so you can draw a conclusion that they are coplanar.