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Prove It $\displaystyle 3x + 4y = -2$
$\displaystyle 4y = -3x - 2$
$\displaystyle y = -\frac{3}{4}x - \frac{1}{2}$.
So for every $\displaystyle 4$ units you have gone to the right, you have gone $\displaystyle 3$ units down.
This would suggest that this line would have direction the same as $\displaystyle 4\mathbf{i} - 3\mathbf{j}$.
But we don't know the length of this vector. So multiply the vector by some parameter $\displaystyle t$.
Therefore the vector that goes in the direction of this line that also has the same length as this line can be written as
$\displaystyle t(4\mathbf{i} - 3\mathbf{j})$.
Now we just need a point that lies on the line so we know where to place this vector. The $\displaystyle y$ intercept will suffice.
Therefore, the vector form of this line is
$\displaystyle \left(0\mathbf{i} - \frac{1}{2}\mathbf{j}\right) + t(4\mathbf{i} - 3\mathbf{j})$
or if you like
$\displaystyle 4t\mathbf{i} - \left(\frac{1}{2} + 3t\right)\mathbf{j}$.