Hi, I've got these 2 questions left on an advanced Mathematics assignment (due Monday morning :( ) that I've been trying to crack but I'm not sure if what I have done is correct. Any help at all is greatly appreciated. Question: (1) (a) According to the Flat Mars Society, Mars is also a plane, given by the equation 4x + 3y − z = −3. Find a parametric equation of the line, L, in which Earth intersects Mars. (5 marks) (b) How far away is Canberra, given by the point (−5, 10, 13), from this line? You are given that the plane describing Earth is given by the equation x + y + z = 18 My Answer: (a) I firstly substituted z = 0 into both plane equations as a point that is on the line where both Earth and Mars intersect, where I got two simultaneous equations: 4x + 3y = -3 x + y = 18 I solved for x and y, and I got the point, P < -57, 75, 0 > that lies on the line of intersection between the two planes. Now, I know that the line that intersects both planes must be perpendicular to both the normal vectors the Earth plane and the normal vector of the Mars plane. Normal Vector(Earth) = < 1, 1, 1 > Normal Vector(Mars) = < 4, 3, -1 > I did the cross product, and ended up getting the vector L = < 4, -5, 1 > Thus, using the values I got from earlier, I wrote them in the form: L = (x - 57)/ 4 = (y + 75)/ -5 = 0 (b) Using the line L = < 4, -5, 1 >, I calculated the distance between L and Canberra D = ( (x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2 )^(1/2) I used the points from L as x1,y1 and z1 and I used the points from Canberra as x2,y2 and z2. After crunching the numbers, I got D = 15((2)^0.5) units as the distance between them. Question 2. Two perfectly round pieces of rock are hurtling through space. The first, Superman’s holiday asteroid, has radius 0.3, and is traveling on the line given by the parametric equation x(t) = 2 + t, y(t) = −1 − t, z(t) = t. The second rock, made of kryptonite and set in motion by one of Superman’s enemies, has radius 0.1 and is traveling on a line given by the parametric equation x(s) = 3 − s, y(s) = 1, z(s) = 1 + s. Calculate the distance between the two lines and use this distance to prove that Superman has nothing to worry about. I attempted this question but I got some reaaaaally obscure answer ( I think I got the answer for the distance between Superman and the Asteroid as 2.39x10^27 :s). Anyway, I've exhausted all my options, reread my lecture notes and the textbook but I can't seem to do this question. Again, ANY help is greatly appreciated. Thanks.