Solving Linear Equation Systems: Intersection & Geometric Interpretation

[kaybaby]

Homework Statement

Determine whether the following planes are parallel or intersect. If they intersect, find the equation of the line of intersection. Interpret this system of two linear equations geometrically.

4x - 3y - z - 1 = 0 and 2x + 4y + z - 5 = 0

Homework Equations

The Attempt at a Solution

I've shown that the plane intersect at one point, and calculated the direction vector, as well as the the two normal vectors. how am i supposed to find the equation of the line of intersection if i do not get a point?

 

[lanedance]

have a think geometrically - how can 2 planes intersect in a single point?

 

[HallsofIvy]

Homework Statement

Determine whether the following planes are parallel or intersect. If they intersect, find the equation of the line of intersection. Interpret this system of two linear equations geometrically.

4x - 3y - z - 1 = 0 and 2x + 4y + z - 5 = 0

These are the same as z= 4x- 3y- 1 and z= -2x- 4y+ 5. On their line of intersection (as lanedance implies, two planes cannot intersect at a single point), z= 4x- 3y- 1= 2x- 4y+ 5. You can solve that for y as a linear function of x, then put that back into either equation to get z as a linear function of x. Set x= t and you have parametric equations for the line of intersection.

Homework Equations

The Attempt at a Solution

I've shown that the plane intersect at one point, and calculated the direction vector, as well as the the two normal vectors. how am i supposed to find the equation of the line of intersection if i do not get a point?