M ASSACHUSETTS I NSTITUTE OF T ECHNOLOGY Interphase Calculus III Worksheet Instructor: Samuel S. Watson 15 July 2015 1. (?) Use vector methods to show that the diagonals of a rhombus are perpendicular. 2. (?) Find the angle formed by two line segments joining a vertex of a regular tetrahedron to its center. (Hint: let the vertices of the regular tetrahedron be (0, 0, 1), (0, 1, 0), (1, 0, 0) and (1, 1, 1), in which case the center would be the average of these four points, namely (1/2, 1/2, 1/2).) 3. Find the distance from the cylinder of radius 2 whose axis is the line given parametrically by (5 + t, −2, 3 + 3t) to the sphere of radius 1 centered at (3, −2, 20). 4. Find equations of the planes that are parallel to the plane x − 2y + 2z = 1 and two units away from it. 5. (?) (a) Find the point at which the lines (1, 1, 0) + t(1, −1, 2) and (2, 0, 2) + s(−1, 1, 0) intersect. (b) Find an equation of the plane that contains these lines. 6. (?) Let L be an invertible linear function from Rn to Rn , and let L−1 be its inverse function. Use the geometric interpretation of | det L| to give a relationship between | det L| and | det L−1 |. 7. (?) For what values of c is the quadric surface x2 + cy2 + (1 + c2 )z2 = 1 unbounded? 8. (?) Sketch the surface defined by the spherical coordinate equation ρ = csc φ. 9. Quickly find the area of the parallelogram shown on the right below. 10. Show that there is no vector v such that (1, 2, 1) × v = (3, 1, 5). 11. (?) Find a formula for the area of a triangle with vertices (0, 0), ( x1 , y1 ), and ( x2 , y2 ). Consider the linear transformation L( x, y) = ( ax + by, cx + dy), where a, b, c, d are constants, and show that the image of any triangle T with one vertex at the origin has area | ad − bc| times the area of T. (Note: this explains why | det L|, defined to be the factor by which areas are scaled under L, is equal to | ad − bc|.) 12. (Challenge problem) Generalize the previous exercise to apply to an arbitrary triangle T. Explain why this implies that the area of the image of any region R under L is | det L| times the area of R. 13. (Challenge problem) Prove that |u × v| = |u||v| sin θ , where θ is the angle between u and v. Hint: begin by expressing |u × v|2 in terms of coordinates and expanding.