Let F = (y,3z + x,3y² ). Use Stokes' Theorem to find a plane with the equation ax + by + cz = 0 (where a, b, c are not all zero) such that , F. dr = 0 for every closed C lying in the plane. %3D Hint: Choose a, b, c so that the curl(F) lies in the plane. (Use symbolic notation and fractions where needed.) Choose the statement that describes a possible equation of the plane. Oc = 3a and b is arbitrary.

Find the cross product a × b. = (t, t², t³), a b = (1, 4t, 9t²)

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)…

Calculate the cross product assuming that u X w = (-2, -3, -2) (2u + 4w) x w = (