I have a question that reads:
Find a parametric equation of each of the following lines:
A. $3 x_1 + 4 x_2 = 6$
D. the line through $A=(-2,1)$ parallel to $x = (1,4) + t(3,5)$
E. the line through $A=(-2,1)$ perpendicular to $x = (1,4) + t(3,5)$
I have been able to solve ones where I am just given two points or the slope and a point, but I don't seem to understand how to find parametric equations when given parallel or perpendicular lines. Could some explain/demonstrate how I would go about this?
In general, in order to write a parametric equation of a line, you need to find one point $A$ of the line and a vector $v$ corresponding to its direction.
A. For the point, we can let $x_2 = 0$, and solve for $x_1$. Then we see that $A = (2, 0)$ belongs to the line. Now, the coordinates of the vector $v = (x_1, x_2)$ must satisfy the homogeneous equation, i.e. $3x_1 + 4x_2 = 0$. So, for example, $v = (4, -3)$ will do. A parametric equation is then $$x = (2, 0) + t (4, -3).$$
D. If the line is parallel to the line of equation $x = (1, 4) + t(3, 5)$, they share the same direction, i.e. you can choose $v = (3, 5)$. Since you are also given a point, you should be able to conclude.
E. If the line is perpendicular to the line of equation $x = (1, 4) + t(3, 5)$, the vector $v = (x_1, x_2)$ is orthogonal to $(3, 5)$, i.e. its coordinates satisfy the equation $3 x_1 + 5 x_2 = 0$. Find a solution for $v$ and write the equation just as before.