East Los Angeles College Calculus 3 Exercise

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novtnvythreen2020

Mathematics

East Los Angeles College

Description

Hi! I need help ASAP with my calculus 3 test.

I only have 15 minutes to submit my answers. Please write clearly,

and use shortcuts when possible. Thank you! 😉

Topics

  • Three-Dimensional Coordinate System
  • Vectors
  • Lines and Planes
  • Surfaces in 3-D
  • I will send a study guide with anwers, maybe this can help you

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    Math 11 Summer 2020 Practice Problems for Quiz 2 (in-class quiz); focus sections 12.2-12.5 Disclaimer: This is a sample of representative problems for extra practice and is not comprehensive. Do not expect the quiz to be completely similar because there are a wide variety of problem types. Be sure that you can do these problems with understanding and without help. 1) Given a = 2, 4, −1 and b = 1, −2,3 , find: (a) 2a − b 2) (b) a (c) a b (d) b ï‚Ž a (e) proja b (f) the angle between a and b Given two orthogonal vectors u and v , and given the length of u is 4 and the length of v is 5, compute the following (clearly explain the properties used and show all steps): 2 (a) u ï‚Ž v (b) u  ( u + v ) (c) u + v 1 3) Use components to prove the following for vectors in V3 . c is a constant. Give reasons for each step. (a) a  a = a 4) 2 (b) a ï‚Ž a = 0 (c) ( ca )  b = c ( a  b ) Let P be a point not on the line L that passes through the points Q and R. Show that the distance d from the point P to the line L is d = a ï‚Žb a where a = QR and b = QP . 2 5) Find parametric equations for the line (a) through ( 4, −1, 2 ) and (1,1,5 ) (b) through ( −2, 2, 4 ) and perpendicular to the plane 2 x − y + 5 z = 12 6) (a) Find a linear equation in three variables for the plane determined by the points P ( −5,3, −3) , Q ( −3, −1,5 ) and R ( 0, 0, −3) . (b) Find the area of PQR . (c) Find the angle of vertex P in PQR . 3 7) Find the point in which the line x = 2 − t , y = 1 + 3t , z = 4t intersects the plane 2 x − y + z = 2 . 8) Find an equation for the plane through (1, 2, −2 ) that contains the line x = 2t , y = 3 − t , z = 1 + 3t 9) Let the plane P contain the line L : r (t ) = 3t , 2 + 6t , 2t − 3 and the point Q (1, −2,3) . Illustrate the following with sketches, and find (a) the distance from Q to L. (use result from #4) (b) a linear equation of the plane P. (c) the equation of the line containing Q and normal to the plane P. (d) the coordinate of the point on line L that is closest to point Q. (this is the projection of Q to L.) 4 10) Given planes P1 : 2 x + y − 2 z = 5 and P2 : 3x − 6 y − 2 z = −15 . Find: (a) the angle of intersection of the planes. (b)the vector equation of the line at which the planes intersect. 5 11) Given two lines L1 : x = 3 + 2t , y = 2 − t , z = −4 + 2t and L2 : x = 5 + 2 s, y = −1, z = s (a) determine if the lines are parallel, skew, or intersecting. If they intersect, find the point of intersection. (b) the equation of the plane containing both lines, if any. 6 12) Given plane P: 3 x + 6 y + 2 z = 6 . Find: (a) the distance from Q ( 2, −5,1) to the plane. (b) the equation of the sphere centered at Q ( 2, −5,1) and tangent to the plane. (c) the point on P closest to Q ( 2, −5,1) . 7 13) Determine the magnitude of the resultant force and its direction measured clockwise from the positive x-axis. Draw the resultant force on the figure using a parallelogram. Concept checks. 14) True or false (a) a ï‚Ž b = b ï‚Ž a (b) −2, 4, −6 is a normal vector to the plane x − 2 y + 3 z − 5 = 0 (c) You can find a plane that contains two skew lines. 8 (d) Two planes parallel to a line are parallel to each other (in space). (e) Two planes each perpendicular to a line are parallel to each other (in space). (f) Two planes either intersect or are parallel. (g) Two lines either intersect or are parallel (in space). (h) A plane and a line either intersect or are parallel. 15) Which of the following are meaningful expressions (meaning you can do the computation)? Answer Yes or No. (a) (a b ) c ( (e) a ï‚Ž b ï‚Ž c ) (b) (a b ) c (c) a b + c (f) (a b )ï‚Ž c (g) a  b ï‚Ž c ( ( ) ) (d) a  b + c (h) a ï‚Ž b 16) Fill in the blanks with the best and most complete answer. (a) a ï‚Ž a = ______ ( (b) a  b ï‚Ž c (c) ) represents the ____________ of a _____________________ . (geometrically) ( a ï‚Ž b )  b = ______ (d) a  a = ______ 9 3) [8] Describe in words and sketch each of the following surface/solid/region, in the 3-dimensional coordinate system. Use the standard orientation for the axes, with the positive z-axis pointing up. (a) y2 = 9 (b) xÂČ + y2 +z?
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    Anonymous
    This is great! Exactly what I wanted.

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