Vector Plane Problem: Find Parametric Eq. of Intersection Line

[Loppyfoot]

Homework Statement

The planes 3y-4x-4z = -18 and 3x-2y+3z = 14 are not parallel, so they must intersect along a line that is common to both of them. The vector parametric equation for this line is: L(t)= ?

Homework Equations

Cross Product Seems like it would be relevant here, but how would I find the intersection point to plug into the result of the cross product?

 

[Mark44]

Homework Statement

The planes 3y-4x-4z = -18 and 3x-2y+3z = 14 are not parallel, so they must intersect along a line that is common to both of them. The vector parametric equation for this line is: L(t)= ?

Homework Equations

Cross Product Seems like it would be relevant here, but how would I find the intersection point to plug into the result of the cross product?

You might be able to do it this way, but it's sort of the long way around. The cross product will give you a vector in the direction of the intersecting line, but you'll still need to find a point that is on the line.

A shorter way would be to solve the system of equations simultaneously. Since there are two equations in three unknowns, there will be an infinite number of solutions. Another way to say this is that the system will have one free variable, which is what you would expect for the solution set to be a line.

You can solve the system simulataneously or you can write the system using an augmented matrix (2 x 4), and use row reduction. When the augmented matrix is row-reduced, write x in terms of z, y in terms of z, and z equaling itself.