atmaca1 - 9 x 2 + 4 y 2 + z 2 = 36 III.Hyperbolic of one...

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Fall 2000 Math22 Section E and G Exam# 1 Murat atmaca 1. Given the vectors a=3i-2k, and b=i-j+k, Find the followings; (a) | a | = (b) (2 a + b ) · (2 a - b ) = (c) The unit vector of a (d) proj a b = (e) The angle θ between the vectors a and b 2. Given a=3i-3j+k and b=i+j-2k. Find a × b and show that a × b is orthogonal to both a and b. 3. Are the planes - 12 x +9 y +3 z = 13 and 8 x - 6 y - 2 z = 4 parallel or perpendicular or neither to each other? 4. For the line passing through the points (1,0,1) and (2,4,7) what must be the values b and c in the following symmetric equation be? x - 1 1 = y - 0 b = z - 1 c . 5. Find the distance between the two skew lines L 1 : x - 1 2 = y - 3 5 = z - 1 3 L 2 : x - 2 4 = y + 1 2 = z + 2 3 . 6. Match each other - x 2 + y 2 - z 2 = 0 I. Ellipsoid. y - x 2 + z 2 = 0 II.Hyperbolic of two sheets.
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Unformatted text preview: 9 x 2 + 4 y 2 + z 2 = 36 III.Hyperbolic of one sheets. x 2-y 2 + z 2 = 0 IV. Hyperbolic Paraboloid. 7. Convert (1,1, 2 ) from the rectangular coordinates to the Spherical coordinates. 8. Given r ( t ) = cos( t ) i + sin( t ) j + e-t k , Find (a) lim t r ( t ) = (b) r ( t ) r ( t ) = (c) Z | r ( t ) | dt = 9. A particle moves with position function r ( t ) = 2 2 ti + e 2 t j + e-2 t k . Find (a) The velocity of r ( t ) at (0,1,1) ; (b) The Acceleration of r ( t ) at (0,1,1); (c) The Speed of r ( t ) at (0,1,1); (d) The Curvature of r ( t ) at (0,1,1); (e) B ( t ) = at (0,1,1)...
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