# Vector equation of a line of intersection of three planes

• May 4th 2010, 12:38 PM
flybynight
Vector equation of a line of intersection of three planes
I have been given a system of equations that appears below:
2x - 7y + 5z = 1
6x + 3y - z = -1
-14x - 23y + 13z = 5
I am told to find an vector equation of the line that is formed by the intersection of these three planes. I could use some step by step instructions, or a link to a resource that explains it well, because as of yet I have found nothing.

Thanks,
Peter
• May 9th 2010, 08:50 PM
dwsmith
Quote:

Originally Posted by flybynight
I have been given a system of equations that appears below:
2x - 7y + 5z = 1
6x + 3y - z = -1
-14x - 23y + 13z = 5
I am told to find an vector equation of the line that is formed by the intersection of these three planes. I could use some step by step instructions, or a link to a resource that explains it well, because as of yet I have found nothing.

Thanks,
Peter

I would start by solving the coefficient matrix. You will end up with that z is a free variable.
• May 10th 2010, 02:32 AM
HallsofIvy
Quote:

Originally Posted by flybynight
I have been given a system of equations that appears below:
2x - 7y + 5z = 1
6x + 3y - z = -1
-14x - 23y + 13z = 5
I am told to find an vector equation of the line that is formed by the intersection of these three planes. I could use some step by step instructions, or a link to a resource that explains it well, because as of yet I have found nothing.

Thanks,
Peter

Is the intersection a line? Normally, two planes intersect in a line so that three planes intersect at a point- with three equations you can solve for specific values of x, y, and z. Presumably, these equations are not "independent". If you don't want to use matrices as dwsmith suggests, try solving the three equations. If they really are not independent, you will find that you cannot solve for x, y, and z but can solve for two of the unknown values in terms ofthe third. Use that write the equation of the line using that third value as parameter.