Q.Show that lines $\vec{r}=(\hat{i}+4\hat{j})+\lambda(\hat{i}+2\hat{j}+3\hat{k})$ and $\vec{r}=(3\hat{j}-\hat{k})+\mu(2\hat{i}+3\hat{j}+4\hat{k})$ 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 via $\vec b_1\times\vec b_2$. …

$\vec a_1=\hat i+4\hat j$, $\vec b_1=\hat i+2\hat j+3\hat k$; $\vec a_2=3\hat j-\hat k$, $\vec b_2=2\hat i+3\hat j+4\hat k$.

$\vec a_2-\vec a_1=(0-1,3-4,-1-0)=(-1,-1,-1)$.

$(\vec a_2-\vec a_1)\cdot(\vec b_1\times\vec b_2)=(-1)(-1)+(-1)(2)+(-1)(-1)=1-2+1=0$, so the lines are coplanar.

Plane: $\vec r\cdot(\vec b_1\times\vec b_2)=\vec a_1\cdot(\vec b_1\times\vec b_2)$. …

Same coplanarity-then-plane routine as the chapter's worked example: verify the scalar triple product of the connecting vector with the two directi …

Mis-reading $\vec a_1=\hat i+4\hat j$ as having a non-zero $\hat k$-component (it has none, since only $\hat i$ and $\hat j$ terms …