Plane intersection

Question

The planes $5x + 3y + 2z = 0$ and $ 2x + 8y - 5z = 0$ intersect. Find the equation of the intersecting line.

I get the parametric equation:

$x = t$

y = $\frac{29}{34}t$

z = $\frac{-121}{170}t$

which I don't think is the right answer. Can someone walk me through the problem?

Answer 1

An algebraic solution: solve the equations simultaneously. If you take the first and then $5$ times the first plus twice the second you get $$5x+3y+2z=0\quad\hbox{and}\quad 29x+31y=0\ .$$ Let $x=t$, solve for $y$ from the second equation, solve for $z$ from the first. As long as your arithmetic is accurate the answer will be right.

A geometric solution: find the normal vector to each plane; the line you want is perpendicular to both these vectors; so its direction can be found by using the cross product. You also need a point on the line, but the origin is clearly on both planes and is therefore a point on their line of intersection.

Answer 2

1. Find a parallel vector to the line of intersection
   $ n_1=(5,3,2),n_2=(2,8,-5)\implies u=n_1×n_2=(-31,29,34)$
2. Find a point of intersection by solving both equation, it is clear both pass through $(0,0,0)$
3. The equation is $r(t)=(0,0,0)+t(-31,29,34)$