Determining Vanishing Points of edges of Triangle using given Point
Vanishing points are "points at infinity" in projective geometry, which are represented by $w = 0$ in homogeneous coordinates. You can construct the vanishing point of a ray or line by taking its $xyz$ direction vector and adjoining $w = 0$. (For a line, use the direction vector and its negation to get vanishing points in both directions.)

Then you can transform this homogeneous vector by your view/projection matrices as usual, and divide out $w$ from the result to find where the point lies in screen space. (If the result has $w \leq 0$ then it's behind the camera or off to the side.)