Mar. 26/19

I. All 3 normals are parallel and distinct - No Intersection.II. 2 normals parallel and distinct, one not - No Intersection.III. All 3 normals parallel and coincident - Intersect in a plane.IV. None parallel and normals not coplanar - Intersect in a point.V. Normals are coplanar but not parallel - Intersect in a line or not at all.

When finding the intersection of three planes:I. Determine if the normals are parallel.II. Determine if the normals are coplanar.Proceed according to the results of I and II.

II. Eliminate the same variable using 2 pairs of equations. Let's eliminate y.

not parallel

1

2 1 6 2 1 63 4 3 3 4 3 2

not coplanarintersect at a point

3

I Determine if parallel and then if coplanar.

2+

Always check your answer

in all the original equations!

the point of intersection is (3,-5, 1).

not parallel1 -5 2 1 -5 21 7 -2 1 7 -2

coplanarintersect in a lineor not at all.

3

II. Eliminate the same variable using 2 pairs of equations. Let's eliminate z.

I Determine if parallel and then if coplanar.

+

2

the planes intersect in the line r = (0, 2, 10) + t(1, -1, -3), t

paralleland distinct

no intersection

1

I Determine if parallel and then if coplanar.

planes are parallel and distinctno intersection

1

I Determine if parallel and then if coplanar.

not parallel

1

1 3 -1 1 3 -12 1 1 2 1 1 2

coplanarintersect ina lineor not at all

3

II. Eliminate the same variable using 2 pairs of equations. Let's eliminate z.

I. Determine if parallel and then if coplanar.

+x 2

+

-

no intersection, planes form a triangular prism.

Pg. 530 # 1, 2, 4, 8ad, 9bc, 13af