MCV4U1 8.5 - The Intersection of 3 PlanesTypes of

Intersection:1.) Parallel Normalsa) Three Parallel and Distinct Planes

b) Two Parallel and Distinct, One is NOT

c) Three Coincident Planes

d) Two Coincident, One is Parallel and Distinct

e) Two Coincident, One is NOT

n1 = jn2 = k n3 And D1 ≠ jD2 ≠ kD3

n1 = jn2 ≠ k n3

And D1 ≠ jD2

D1 = jD2 = kD3

D1 = jD2 ≠ kD3

D1 = jD2 And

And

Andn1 = jn2 = k n3

n1 = jn2 = k n3

n1 = jn2 ≠ k n3

***Find the line of intersection using Gaussian Elimination***

Types of Intersection:2.) Non-Parallel Normalsa) Intersect at a Point

b) Intersect along a line

c) Triangular Prism

***Solve using Gaussian Elimination***

***Solve using Gaussian Elimination*** Bottom row results in [0 0 0 0]

***Solve using Gaussian Elimination***

Bottom row results in [0 0 0 #]

a) x + 2y + 3z = -4 2x + 4y + 6z = 7 x + 3y + 2z = -3

b) x + 2y + 3z = -4 2x + 4y + 6z = 7 3x + 6y + 9z = 5

c) x + 3y - z = -1 - x - 3y + z - 1 = 0 4x + 12y - 4z = -4

d) x + 2y + 3z = -4 x - y - 3z - 8 = 0 2x + y + 6z + 14 = 0

Ex.) Determine the intersection of the following sets of planes. Provide a geometric interpretation for each example.

e) x + 2y + 3z = -4 x - y - 3z = 8 x + 5y + 9z = -16

f) x + 2y + 3z = -4 x - y - 3z = 8 x + 5y + 9z = -10

Ex.) Determine the intersection of the following sets of planes. Provide a geometric interpretation for each example.

Homework: p. 309 # 7, 8, 9

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