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# Math280 Math280

Montgomery College

GPA 4.0

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This 4 page Study Guide was uploaded by Nhi Phan on Wednesday June 15, 2016. The Study Guide belongs to Math280 at Montgomery College taught by Dr.Muse in Summer 2016. Since its upload, it has received 16 views. For similar materials see Multivariate Calculus in Mathematics (M) at Montgomery College.

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Date Created: 06/15/16

MATH 280- Worksheet I –Summer 2016 –Instructor: Dr. Muse kahsay 1. Which of the following statements are true? a. ( )and ( ) lie on the same side of the plane . b. The set of all points that are equidistant from ( ) and ( ) is a plane. c. The triangle with vertices ( ), ( ) and ( ) is a right-angled triangle. d. The acute angle between the lines with equations and is . e. The scalar projection of onto is√ f. The vector 〈 〉 is orthogonal to both 〈 〉 and 〈 〉. g. The line through ( ) and ( ) is perpendicular to the line through ( )and ( ). h. The two planes with equations and are perpendicular to each other. i. The equation represents an elliptic paraboloid. 2. Find a unit vector with same direction as but has length 7. 3. Find two unit vectors that make an angle 60 with 〈 〉. Hint: Find a system of nonlinear equations and solve it using Wolfram Alpha. 4. Find a vector equation and parametric equation of the line through ( ) and ( ). At what point does this line intersect the plane ? 5. Suppose we are given three points ( ) ( ) and ( ). a. Let be the angle between ⃗⃗⃗⃗and ⃗⃗⃗. Find . b. Find a vector perpendicular to the plane through the points , , and . c. Use your answer for question b to find the area of the triangle . d. Use b to find the equation of the plane through , and . e. Show that (⃗⃗⃗⃗⃗⃗⃗⃗,⃗⃗⃗) if ( ) is a point on the plane through , and .Give a geometric interpretation. 6. Find the angle between the planes and . Find symmetric equations of line of intersection of these two planes. 7. Find the volume of the parallelepiped with adjacent edges , , and , where ( ) ( ), ( ) and ( ). Do these points lie on the same plane? 8. Use determinants to find the area of the parallelogram with vertices a. ( ), ( ), ( ), and ( ). b. ( ), ( ), ( ), and ( ). 1 of 4 9. Use scalar projections to find the distance from the point ( ) to the a. plane through ( ) , ( ) and ( ) b. line through ( ) and ( ). 10. Find the distance between two parallel planes and . 11. Show that lines with parametric equations , , , and , are skew, and find the distance between them. 12. a. Show that the equation represents a sphere, and find its center and radius. b. Find an equation of the sphere with center ( ) that touches the -plane. c. Find an equation of the largest sphere with center ( ) that is contained in the first octant. d. Find an equation of the sphere that passes through the point ( ) and has center( ). 13. Two forces and of magnitudes 12 lb and 10lb act on an object at a point as shown in the figure. Find the resultant force acting at as well as its magnitude and its direction.(Indicate the direction by finding the angle shown in the figure.) 14. A women walks due west on the deck of a ship at 3mi/h. The ship is moving north at a speed of 22 mi/h. Find the speed and direction of the women relative to the surface of the water. 15. Ropes 3 m and 5 m in length are fastened to an object that is suspended over a town square. The object has a weight of 7 kg. The ropes, fastened at different heights, make angles of 52° and 40° with the horizontal. Call the two tensile forces and , corresponding to the ropes of length 3 m and 5 m. Find and and give your answers correct to two decimal places. 2 of 4 16. A constant force moves an object along the line segment from the point P(1,0,2) to the point Q(5,3,8). Find the work done if the distance is measured in meters and the force in newtons. 17. Find the domain of each of the following functions. a. ( ) √ b. ( ) ( ) √ c. ( ) 18. Match the surface with its name. Select all that apply. 19. Write the description the solid region in terms of inequalities involving spherical coordinates. a. Above the cone √ and below the plane . b. Above the cone √ and below the sphere . c. Above the cone √ and below the sphere . 20. Write the description the solid region given in question 19 in terms of inequalities involving cylindrical coordinates. 3 of 4 Answers 1. b, c, d, e, f , h & i 2. √ √ √ 10. √ 3. 〈 ( √ ) ( √ )〉 11. √ and 〈 ( √ ) ( √ )〉 12. 4. ( ) ( ) ; a. Center: ( ) and . b. ( ) ( ) ( ) . , and c. ( ) ( ) ( ) ( ) d. ( ) ( ) ( ) 5. a. ( √ ) 13. ( √ √ ) ( √ ) and b. 〈 〉 √ ( √ √ ) . c. d. 14. √ ; 15. and e. The volume of the parallelepiped determined ⃗⃗⃗⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗ by the vectors ,& is zero. That is 16. 87 J lie on the same plane. 17. 6. ( ) a. {( ) } and b. {( ) } 7. 0; yes c. {( ) } 8. 18. Do it. a. 16 19. b. √ a. and 9. b. and . a. c. and . √ b. √ 20. a. and b. √ and √ c. and √ 4 of 4

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