Q.Find the vector equation of the plane passing through the origin and containing the line $\vec{r}=(\hat{i}+4\hat{j}+\hat{k})+\lambda(\hat{i}+2\hat{j}+\hat{k})$.

[!TLDR] Plane through origin containing a line: normal $=$ (point on line) $\times$ (line's direction). …

The given line $\vec r=(\hat i+4\hat j+\hat k)+\lambda(\hat i+2\hat j+\hat k)$ passes through $P(1,4,1)$ with direction $\vec b=(1,2,1)$. A plane through the origin containing this line must also contain the vector $\vec{OP}=(1,4,1)$ (from the origin to any point on the line) as well as the line's own direction $\vec b$. …

A plane through the origin containing a given line is spanned by the position vector of any point on the line and the line's own direction vector; cr …

Using the line's direction vector twice instead of pairing it with the point-on-line vector (the plane needs two genuinely different vectors spanning it — the ori …