Q.Find the vector equation of the line passing through the point $(1,2,3)$ and parallel to the vector $2\hat i-\hat j+2\hat k$. Also write its Cartesian equation.

With $\vec a=\hat i+2\hat j+3\hat k$ (position vector of $(1,2,3)$) and $\vec b=2\hat i-\hat j+2\hat k$, the vector equation is

Writing $\vec r=x\hat i+y\hat j+z\hat k$ and equating components: $x=1+2\lambda,\ y=2-\lambda,\ z=3+2\lambda$. Solving each for $\lambda$ gives the Cartesian equation …

Substitute the given point and direction vector directly into $\vec r=\vec a+\lambda\vec b$; equate $\hat i,\hat j,\hat k$ components and s …

Writing the Cartesian denominator for $y$ as $1$ instead of $-1$ (dropping the sign from $-\hat j$); mixing up the point' …