What is the vector equation of the line $2x - 4 = 3y + 3 = 6 - z$?

Explanation

1. 01

   Take out the coefficients: $2(x - 2) = 3(y + 1) = -(z - 6)$; dividing by 6, $\frac{x - 2}{3} = \frac{y + 1}{2} = \frac{z - 6}{-6}$.

2. 02

   So the line passes through $(2, -1, 6)$ with direction $3\hat{i} + 2\hat{j} - 6\hat{k}$: $\vec{r} = (2\hat{i} - \hat{j} + 6\hat{k}) + \lambda(3\hat{i} + 2\hat{j} - 6\hat{k})$.