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

You are given two planes in parametric form,

1:x1x2x3=333+1111+22301:(x1x2x3)=(333)+1(111)+2(230)

2:x1x2x3=313+1111+22102:(x1x2x3)=(313)+1(111)+2(210),

where x1,x2,x3,1,2,1,2Rx1,x2,x3,1,2,1,2R. Let LL be the line of intersection of 11 and 22.

1. Find vectors n1n1 and n2n2 that are normals to 11 and 22 respectively and explain how you can tell without performing any extra calculations that 11 and 22 must intersect in a line.
2. Find Cartesian equations for 11 and 22.
3. For your first method, assign one of x1x1, x2x2 or x3x3 to be the parameter and then use your two Cartesian equations for 11 and 22 to express the other two variables in terms of and hence write down a parametric vector form of the line of intersection LL.
4. For your second method, substitute expressions for x1x1, x2x2 and x3x3 from the parametric form of 22 into your Cartesian equation for 11 and hence find a parametric vector form of the line of intersection LL.
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=n1n2m=n1n2 and show that mm is parallel to the line you found in parts (c) and (d).
7. Give a geometric explanation of the result in part (f)

You can make some checks to ensure you are on the right track with your calculations.

1. A possible n1n1 is (Use Maple syntax, eg <1, 2, 3>.) A possible n2n2 is (Use Maple syntax, eg <1, 2, 3>.)
2. Enter your Cartesian equation for 11 here: (Use the variable x1, x2 and x3.) Enter your Cartesian equation for 22 here: (Use the variable x1, x2 and x3.)
3. A parametric equation of the line will have the form x=a+vx=a+v for RR. Check your values for aa and vv. Your a=a= (Use Maple syntax, eg <1, 2, 3>.) Your v=v= (Use Maple syntax, eg <1, 2, 3>.)
4. You can reuse the answer boxes in the check for part (c) to verify your second parametric form for the line if it is different.
5. You're on your own here :-)
6. You can check your own mm. Remember that the cross product of two vectors is perpendicular to those vectors.
7. You're on your own here :-)