Question

In this question you will find the intersection of two planes using two different methods.

You are given two planes in parametric form,

1. Find vectors n1 and n2 that are normals to Π1 and Π2 respectively and explain how you can tell without performing any extra calculations that Π1 and Π2 must intersect in a line.
2. Find Cartesian equations for Π1 and Π2.
3. For your first method, assign one of x1, x2 or x3 to be the parameter ω and then use your two Cartesian equations for Π1 and Π2 to express the other two variables in terms of ω and hence write down a parametric vector form of the line of intersection L.
4. For your second method, substitute expressions for x1, x2 and x3 from the parametric form of Π2 into your Cartesian equation for Π1 and hence find a parametric vector form of the line of intersection L.
5. If your parametric forms in parts (c) and (d) are different, check that they represent the same line. If your parametric forms in parts (c) and (d) are the same, explain how they could have been different while still describing the same line.
6. Find m=n1×n2 and show that m is parallel to the line you found in parts (c) and (d).
7. Give a geometric explanation of the result in part (f)

Answer 1