a. The vector component of E at P (5,rad, 3) parallel to the line x = 2, z = 3. b. The angle E makes with the surface z = 3 at the point P.

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)…

2. Let v be a vector parallel to a line that passing through the points P(1,0,-3) and Q(3,3,-3). a. Find the equation of the plane that passing through P and has v as its normal vector. b. Find the distance of that plane to the Q.

a) Let the 3-D vector a make angles a, ß and y with the coordinate axes. Show that â = cos ai + cos Bj + cos yk is a unit vector in the direction of the vector a. Show also that cos²a + cos?B + cos²y = 1. b) The path of a particle is given by the position vector 7(t) = a sin ti + a cos tj + b tk, where a and b are real constants. i) Sketch the graph of the curve traced by the particle as t varies. ii) Find the length of the curve between the points correspondong to t = 0 and t = 27. c) The displacement of a particle in space is given by the vector r'(t) = sint cost i+sin²t }+cost k. Find the tangent, the normal and binormal, and the curvature and torsion of the curve at the points (i) t= 0 and (ii) t = "/2.