(b) A surface, S, is given by f(r) = x²y + y²z + z²x = 4. Confirm that the point with position vector a lies on both the surface S and the plane P and find the angle of intersection between S and P at that point. Hint: consider the normals to the surface and plane at this point.

Does the sphere x2+y2+z2=100 have symmetry with respect to the a x-axis? b xy-plane?

Find an equation of the plane through the three points given: Р - (0,5, 0),Q — (2, 1, —4), R — (-1,1, —3) P = 50

You are given two planes in parametric form, x1 Il : x2 X1 IL : x2=|0 +41 2 + 2 4 X3 where X1 , X2 , X3, 11 , d2,41, H2 E R. Let L be the line of intersection of п and IL. a. Find vectors nj and nɔ that are normals to П and IL respectively and explain how you can tell without performing any extra calculations that II and II, must intersect in a line. b. Find Cartesian equations for П and IL. c. For your first method, assign one of X2 or X3 to be the parameter o and then use your two Cartesian equations for П and II, to express the other two variables in terms of o and hence write down a parametric vector form of the line of intersection L. d. For your second method, substitute expressions for X1 » X and X3 from the parametric form of II, into your Cartesian equation for II, and hence find a parametric vector form of the line of intersection L. e. 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)…