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chuang_1

# chuang_1 - 0(a(5 pts Find the cosine of the angle between...

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MATH 103: EXAM 1 Time: 50 minutes. No calculators allowed. (1) (35 pts) Let P = (2 , 4 , 5) Q = (1 , 5 , 7) R = ( - 1 , 6 , 8) be points in R 3 . (a) (10 pts) Find an equation for the plane containing points P, Q, R . (b) (5 pts) Find the area of the triangle determined by P, Q, R . (c) (10 pts) Find an equation for the line determined by the intersection of the plane x + y + z = 0 with that in (a). (d) (10 pts) Compute the distance between point R and the line deter- mined by points P, Q . (2) (30 pts) Tschirnhausen’s cubic is the plane curve described by the para- metric equations: x ( t ) = 3( t 2 - 3) y ( t ) = t ( t 2 - 3) This curve intersects itself only once, at the point (0
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Unformatted text preview: , 0). (a) (5 pts) Find the cosine of the angle between the two tangent vectors at the self-intersection (0 , 0). (b) (10 pts) Find the arc-length of the loop created. (c) (10 pts) Find an expression for the curvature κ . (d) (5 pts) Compute lim t →∞ κ . (3) (5 pts) In triangle ABC , let P be the midpoint of side BC . Show if P is equidistant from vertices A, B, C , then the angle at A is a right angle. Important: Please copy and sign the following Reaﬃrmation: I have adhered to the Duke Community Standard in completing this assignment. 1...
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