Calculus 3: Equations of lines and planes

Find the vector equation, parametric equations, and symmetric equations for the line that passes through the points $(1, 3, -2)$ and $(4, 1, 5)$.

Find the equation of a plane given the three points P(2, 1, 4), Q(4, -2, 7), and R(5, 3, -2).

Given a point $P_0 = (1, 2, 3)$ and a normal vector $\mathbf{n} = (4, 5, 6)$, find the equation of the plane in component form.

Find the parametric and symmetric equations of a line in space given two points.

Parametrize the plane given by the equation $2x - 3y + z = 6$ using two parameters $u$ and $v$.