Q.Show that lines $\vec{r}=(2\hat{j}-3\hat{k})+\lambda(\hat{i}+2\hat{j}+3\hat{k})$ and $\vec{r}=(2\hat{i}+6\hat{j}+3\hat{k})+\mu(2\hat{i}+3\hat{j}+4\hat{k})$ are coplanar. Find the equation of the plane determined by them.

[!TLDR] Coplanar iff $(\vec a_2-\vec a_1)\cdot(\vec b_1\times\vec b_2)=0$; then plane is $(\vec r-\vec a_1)\cdot(\vec b_1\times\vec b_2)=0$.

$\vec a_1=2\hat j-3\hat k$, $\vec a_2=2\hat i+6\hat j+3\hat k$, $\vec b_1=\hat i+2\hat j+3\hat k$, $\vec b_2=2\hat i+3\hat j+4\hat k$; the s …

Here $\vec a_1=2\hat j-3\hat k=(0,2,-3)$, $\vec b_1=\hat i+2\hat j+3\hat k=(1,2,3)$, $\vec a_2=2\hat i+6\hat j+3\hat k=(2,6,3)$, $\vec b_2=2\hat i+3\hat j+4\hat k=(2,3,4)$.

$\vec a_2-\vec a_1=(2,4,6)$.

Since this is $0$, the two lines are coplanar.

Now $\vec b_1\times\vec b_2=\begin{vmatrix}\hat i&\hat j&\hat k\\1&2&3\\2&3&4\end{vmatrix}=\hat i(8-9)-\hat j(4-6)+\hat k(3-4)=-\hat i+2\hat j-\hat k$. …

Identify $\vec a_1,\vec b_1,\vec a_2,\vec b_2$ from the two vector line equations, verify $(\vec a_2-\vec a_1)\cdot(\vec b_1\times\vec b_2)=0$ for coplanarity, then use $\vec b_1\times\vec b_2$ as t …

Swapping which vector is $\vec a_1$ and which is $\vec a_2$ when forming the difference (this does not affect the coplanarity zero-check but does flip the sign of the plane's constant if no …