   Chapter 10.6, Problem 46E Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085

Solutions

Chapter
Section Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085
Textbook Problem

The planes with the equation x + 3 z = 12 and y - 5 z = 10 intersect in a line. Find the equation for the line in the form x ,   y ,   z = x 1 ,   y 1 ,   z 1 + a ,   b ,   c .

To determine

To explain:

Find the equation for the line of intersection in the form x, y, z=x1, y1, z1+a, b, c.

Explanation

The plane equations are x+3z=12 and y-5z=10.

The planes have normal vectors.

For the first plane the normal vector a=1, 0, 3

For the second plane the normal vector b=0, 1, -5

Let L be the line of intersection of the two planes.

Now, we have to find a×b.

a×b is parallel to L.

a×b=ijk10301-5

a×b=-3i--5j+1k

This can be simplified as,

a×b=-3i+5j+1k

This is the coordinates of the direction vector

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