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OLD MATHEMATICS 23 EXAMINATION 1

# OLD MATHEMATICS 23 EXAMINATION 1 - the point(0 0 2 Simplify...

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OLD MATHEMATICS 23 EXAMINATION 1 Reminder: The actual examination may be very di_erent. (1) Let u = i_j+2k and let v = 2i_j+k. Are the vectors u and v orthogonal, parallel, or neither? Explain carefully. (2) Find an equation of the plane P containing the points (3;_1; 2), (8; 2; 4) and (_1;_2;_3). (3) Let P be the plane in the above problem. Find a parametric equation for the line through (_2; 2; 4) and perpendicular to the plane P. (4) Carefully sketch the surface 4y = x 2 + z 2 . (5) Describe the surface S given by the spherical coordinate equation _ sin ' = 2 in terms of an equation involving rectangular coordinates x; y; z. Identify the surface S. (6) let r(t) = h2 sin t; 5t; cos ti. Find the curvature of the curve r(t) at
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Unformatted text preview: the point (0; 0; 2). Simplify your answer. (7) Consider the helix r(t) = h2 cos t; sin t; ti where t 2 (_1;1). Fine a parametric equation of the tangent line to the helix at the point _0; 1; _ 2 _. (8) Find the length of the space curve f (t) = 12ti + 8t 3=2 j + 3t 2 k, 0 _ t _ 1. Your numerical answer should be in simplest possible form. (9) Find the space curve r(t), t 2 [a; b], which represents the curve of intersection of the cylinder x 2 + y 2 = 4 and the plane y + z = 3. You will get some partial credit for correctly sketching the surfaces. You need to _nd both r(t) and the interval [a; b]. 1...
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