Give a geometric description of the set of points in space whose coordinates satisfy the given pair of equations.
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1 1. Give a geometric description of the set of points in space whose coordinates satisfy the given pair of equations. x + y = 5, z = 4 Choose the correct description. A. The circle with center (0,0, 4) and radius 5, parallel to the xy plane B. The circle with center ( 5, 5,0) and radius 4, parallel to the xy plane C. The line that passes through the points ( 5,0, 4) and (0, 5, 4) D. The line through (5,5, 4), parallel to the z axis Answer: A. The circle with center (0,0, 4) and radius 5, parallel to the xy plane ID: Give a geometric description of the set of points in space whose coordinates satisfy the given pairs of inequalities. a. z x, y 0 b. x y, 0 z a. Choose the correct description. A. The region on or inside the parabola z = x in the xz plane and all points to the left of this region B. The region on or inside the parabola z = x in the xz plane and all points to the right of this region C. The region on or outside the parabola z = x in the xz plane and all points to the right of this region D. The region on or outside the parabola z = x in the xz plane and all points to the left of this region b. Choose the correct description. A. The region on or inside the parabola x = y in the xy plane and all points above it which are units or less away from the xy plane B. The region on or outside the parabola x = y in the xy plane and all points above it which are units or less away from the xy plane C. The region on or inside the parabola x = y in the xy plane and all points below it which are units or less away from the xy plane D. The region on or outside the parabola x = y in the xy plane and all points below it which are units or less away from the xy plane Answers C. The region on or outside the parabola z = x in the xz plane and all points to the right of this region B. The region on or outside the parabola x = y in the xy plane and all points above it which are units or less away from the xy plane ID: 1.1.
2 . Describe the plane perpendicular to each of the following axes at the given points with a single equation or a pair of equations. a. the z axis at (,9,4) b. the x axis at (8, 1,0) c. the y axis at (8,, ) a. Choose the correct equation of a plane perpendicular to the z axis. A. y = 9 B. x = C. z = 4 b. Choose the correct equation of a plane perpendicular to the x axis. A. x = 8 B. y = 1 C. z = 0 c. Choose the correct equation of a plane perpendicular to the y axis. A. z = B. y = C. x = 8 Answers C. z = 4 A. x = 8 B. y = ID: Find the distance between points P 1 and P. P 1(4,1,), P (10,,5) The distance is. Answer: ID: 1.1.4
3 5. Determine the equation for the sphere whose center and radius are given. Center Radius (4,4,5) Choose the correct equation of the sphere. A. (x 4) + (y 4) + (z 5) = B. (x + 4) + (y + 4) + (z + 5) = C. (x 4) + (y 4) + (z 5) = D. (x + 4) + (y + 4) + (z + 5) = Answer: C. (x 4) + (y 4) + (z 5) = ID: Find the center and radius of the sphere. x + y + z + x + y + z = 1 Center = (,, ), radius = Answers ID: Let u = 4, 5 and v = 1, 4. Express u + v in the form a,b. u + v =, (Simplify your answers.) Answers 9 ID: 1..
4 8. Find the component form of the vector from the point A = (5,1) to the origin. The component form of the vector from the point A = (5,1) to the origin is,. (Simplify your answers.) Answers 5 1 ID: 1..11
5 9. Copy vectors u, v, and w head to tail as needed to sketch the indicated vectors. a. u + v b. u + v + w c. u v d. u w v w u Choose the correct sketch for u + v. A. B. C. D. Choose the correct sketch for u + v + w. A. B. C. D. Choose the correct sketch for u v. A. B. C. D. Choose the correct sketch for u w. A. B. C. D.
6 Answers D. A. B. D. ID: Express the vector i + 6 j + 6 k as a product of its length and direction. i + 6 j + 6 k = [( ) i + ( ) j + ( ) k] (Simplify your answers. Use integers or fractions for any numbers in the expression.) Answers ID: 1..5
7 11. Find the vectors whose lengths and directions are given. Try to do the calculations without writing. Length a. 6 i b. i c. 1 Direction 4 j k 5 5 d. i j k + 6 a. The vector with length 6 and direction i is ( ) i + ( ) j + ( ) k. b. The vector with length and direction i is ( ) i + ( ) j + ( ) k. 1 4 c. The vector with length and direction j k is 5 ( 5 ) i + ( ) j + ( ) k. d. The vector with length and direction i j k is + 6 ( ) i ( ) j + ( ) k. Answers ID: 1..1
8 1. For the points P 1(,,) and P (4,4,1), find the direction of P 1 P and the midpoint of line segment P 1 P. The direction of P 1 P is ( ) i + ( ) j + ( ) k. The midpoint of P 1 P is (,, ). (Type integers or simplified fractions.) Answers ID: 1..
9 1. Find the following for the vectors u = 8 i 9 j + k and v = 8 i + 9 j k. a. v u, v, and u b. the cosine of the angle between v and u c. the scalar component of u in the direction of v d. the vector proj v u v u = (Simplify your answer.) v = (Type an exact answer, using radicals as needed.) u = (Type an exact answer, using radicals as needed.) The cosine of the angle between v and u is. (Type an exact answer, using radicals as needed.) The scalar component of u in the direction of v is. (Type an exact answer, using radicals as needed.) proj v u = ( ) i + ( ) j + ( ) k Answers ID: 1..1
10 14. Given the vectors v and u, answer a. through d. below. v = 11 i + j 10 k u = i + 4j a. Find the dot product of v and u. u v = Find the length of v. v = (Simplify your answer. Type an exact answer, using radicals as needed.) Find the length of u. u = (Simplify your answer. Type an exact answer, using radicals as needed.) b. Find the cosine of the angle between v and u. cos θ = (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) c. Find the scalar component of u in the direction of v. (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) d. Find the vector projection of u onto v. proj v u = (Type your answer in terms of i, j, and k. Use integers or fractions for any numbers in the expression. Do not factor.) Answers i + j k ID: Find the angle between the vectors u = i j and v = i + j k. The angle between the vectors is θ radians. (Do not round until the final answer. Then round to the nearest hundredth as needed.) Answer: 1.66 ID: 1..11
11 16. Find the measures of the angles of the triangle whose vertices are A = ( 1,0), B = (1,1), and C = (, ). (Round to the nearest thousandth.) The measure of ABC is. (Round to the nearest thousandth.) The measure of BCA is. (Round to the nearest thousandth.) The measure of CAB is. Answers ID: 1..1
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