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Unformatted text preview: Math 2110Q Vectors and Vector Functions V1.0  Spring  2010 1. Normal plane
(6 points)
Find an equation of the normal plane to the curve r(t) = 2t, e2t , sin(π t) at the point (2, e2 , 0). ANSWER 2. Tangent line
(4 points)
2 , ln t, 7 when t = 5.
Find a vector equation of the tangent line to the curve r(t) = t ANSWER 1 Math 2110Q Vectors and Vector Functions V1.0  Spring  2010 3. Vector algebra
Find all values of x such that the vectors −3, 2, x and 2x, 4, x are orthogonal. (2 points) ANSWER
x= 4. Vector algebra
If a = 4i − 2j and b = −2j − k, calculate a × b and proja b. (6 points) ANSWER
a×b=
proja b = 2 Math 2110Q Vectors and Vector Functions V1.0  Spring  2010 5. Curvature:
Find the curvature of r(t) = cos t, sin t, t2 at the point (1, 0, 0). (8 points) ANSWER
κ= 3 Math 2110Q Vectors and Vector Functions V1.0  Spring  2010 6. Intersection of two surfaces:
(8 points)
Find a vector function that represents the curve of intersection of the surfaces z = 2x2 + 3y 4
and x = y 2 + 1. ANSWER 4 Math 2110Q Vectors and Vector Functions V1.0  Spring  2010 7. Intersection of two curves: (8 points) (a) Find the point of intersection of the two curves r1 (t) = t + 1, t, 4 + 2t + t2 and
r2 (s) = 3 − s, s − 2, s2 , checking that the point really lies on both curves. ANSWER
(x, y, z ) =
(b) Find the angle of intersection between the two curves in part (a). ANSWER
θ= 5 Math 2110Q Vectors and Vector Functions V1.0  Spring  2010 8. Quadratic surfaces:
(8 points)
You do not need to make any sketches as part of your answers on this page (though you are
welcome to if it helps you). You do need to justify your answers mathematically, as always.
(a) What type of conics sections are the traces of z = 4x2 + 3y 2 in the planes y = k ? (Your
ﬁnal answer should be one word which tells me the shape.) ANSWER (b) What type of conics sections are the traces of x2
2 + y2
5 − z 2 = 1 in the planes x = k ? ANSWER (c) Find the center C of the sphere x2 + 2y 2 + z 2 + 12y + 17 = 0. ANSWER
C=
(d) For what values of k is the trace of −x2 + y 2 − z 2 = 4 in the plane y = k nonempty? ANSWER 6 ...
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 Fall '07
 BoonYeap

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