Q.Find the vector and Cartesian equations of the line passing through the point $(2,-3,4)$ and parallel to the vector $\hat i+2\hat j-2\hat k$.

Vector equation: $\vec r=(2\hat i-3\hat j+4\hat k)+\lambda(\hat i+2\hat j-2\hat k)$.

Cartesian equation: $\dfrac{x-2}{1}=\dfrac{y-(-3)}{2}=\dfrac{z-4}{-2}$, i.e. $\dfrac{x-2}{1}=\dfrac{y+3}{2}=\dfrac{z-4}{-2}$.

Read the point's position vector and the given direction vector directly into $\vec r=\vec a+\lambda\vec b$, then equate components for the Cartesian symmetric form.

Writing $(y-3)$ instead of $(y+3)$ from the negative $y$-coordinate; dropping the negative sign on the $z$-direction ratio.