Find the co-ordinates of the point on the join of two points which is nearest to the intersection of two planes

Question

Find the co-ordinates of the point on the join of $(-3, 7, -13)$ and $(-6, 1, -10)$ which is nearest to the intersection of the planes $3x-y- 3z + 32 =0$ and $3x+2y-15z= 8$.

Please give me an outline to solve the problem. Thanks.

Answer 1

You do not tell us what mathematics you already know. Here is one outline, which is not clever but is straightforward. This method avoids calculus, since you did not use that tag, but it does use vectors. Let us know if one of these steps is too difficult for you.

1. Find the vector $v_1$ between those two given points $A_0,A_1$ and the parametric equations of the line between them $l_1$.
2. Solve the two given equations for the planes and find the parametric equations of the line of intersection $l_2$ and a direction vector $v_2$ for that line.
3. Find a vector $v$ perpendicular to the two vectors you just found $v_1,v_2$.
4. For each of your two lines, use the perpendicular vector to find the equation of the plane $p_1,p_2$ through the line that contains the vector. Then find the point of intersection $C_1,C_2$ of that plane and the other line.
5. See if the point of intersection $C_1$ on your first line is between the two given points $A_0,A_1$. If it is, that is your desired "nearest point." If it is not, find the distance between each of your two given points $A_0,A_1$ and the point $C_2$ on the other line $l_2$. The closer one of $A_0,A_1$ is your desired nearest point.