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 $(4, 1, 5)$.

Given a fixed point $P_0$ with coordinates $(x_0, y_0, z_0)$ and a direction vector $\vec{v}$ in three-dimensional space, find the vector equation of a line that passes through $P_0$ and is parallel to $\vec{v}$.

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 a vector equation and parametric equations for the line that passes through the point $(5, 1, 3)$ and is parallel to the vector $\mathbf{v} = (1, 4, -2)$. Then find two other points on the line.

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 $v$.