Sect.12.3 Enhanced Assignment

Instructor: Shijun Zheng Course: Spring 2023 - Calculus III (MATH2243-A-B) Student: Jalen Wilson Date: 07/11/24 1. Find the following for the vectors u = 10i + 5j + 2 k and v = − 10i − 5j − 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 projv u Assignment: Sect.12.3 Enhanced Assign 2 k. v•u= − 127 ​(Simplify your​answer.) v = 127 ​(Type an exact​answer, using radicals as needed.) u = 127 ​(Type an exact​answer, using radicals as needed.) The cosine of the angle between v and u is −1 ​(Type an exact​answer, using radicals as needed.) The scalar component of u in the direction of v is . − 127 . ​(Type an exact​answer, using radicals as needed.) projv u = 10 i+ 5 j+ 2 k ​(Type exact​answers, using radicals as​needed.) 2. Given the vectors v and u​, answer a. through d. below. v = 11i + 2j − 10k u = 7i + 24j a. Find the dot product of v and u. u•v= 125 Find the length of v. v = 15 ​(Simplify your answer. Type an exact​answer, using radicals as​needed.) Find the length of u. u = 25 ​(Simplify your answer. Type an exact​answer, using radicals as​needed.) b. Find the cosine of the angle between v and u. 1 cos θ = 3 ​(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. 25 3 ​(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. projv u = 55 9 i+ 10 9 j+ − 50 9 k ​(Type your answer in terms of​i, j, and k. Use integers or fractions for any numbers in the expression. Do not​factor.) 3. Find the angle between the vectors u = − 4i + 5j and v = − 3i + 6j − 5k. The angle between the vectors is θ ≈ ​(Round to the nearest​hundredth.) 0.67 radians. 4. Solve the triangle. 80° c a = 13.1 γ 77° b γ= °​, b = 23 13.2 ​, c = 5.3 ​(Round to the nearest tenth as​needed.) 5. Verify that the trigonometric equation is an identity. 1 − cos x 1 + cos x − 1 + cos x 1 − cos x = − 4 cot x csc x Which of the following statements establishes the​identity? A. 1 − cos x 1 + cos x B. 1 − cos x 1 + cos x C. 1 − cos x 1 + cos x D. 1 − cos x 1 + cos x − − − 1 + cos x 1 − cos x 1 + cos x 1 − cos x 1 + cos x 1 − cos x − 2 = ( csc x − 1) − ( csc x + 1) 2 ( csc x + 1)( csc x − 1) (1 − cos x) − (1 + cos x) 1 + cos x 1 − cos x 2 ( sec x − 1) − ( sec x + 1) 2 4 csc x (1 − sin x) − (1 + sin x) 2 − 4 sec x = 2 =− 4 cos x 2 =− (1 + sin x)(1 − sin x) 4 sec x 2 = − 4 cot x csc x tan x − 4 sin x = = − 4 cot x csc x sin x sec x − 1 2 = − 4 cot x csc x 2 cot x 1 − cos x 2 ( sec x + 1)( sec x − 1) = =− − 4 cos x = (1 + cos x)(1 − cos x) 2 = 2 csc x − 1 2 = − 4 csc x = =− 2 1 − sin x 4 sin x 2 = − 4 cot x csc x cos x 6. The parametric equations and parameter intervals for the motion of a particle in the​xy-plane are given below. Identify the​particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. Indicate the portion of the graph traced by the particle and the direction of motion. x = cos (2t)​, y = sin (2t)​, 0 ≤ t ≤ π The Cartesian equation for the particle is 2 2 x +y =1 . Choose the correct graph that represents this motion. A. B. 6 y C. 6 y 6 x -6 6 D. y 6 x -6 -6 6 -6 y x -6 6 -6 x -6 6 -6 7. Find a parameterization for the circle (x − 18)2 + y2 = 81 starting at the point ​(9​,0) and moving clockwise once around the circle. Find parametric equations for the circle. x= 18 − 9 cos θ ​, y = 9 sin θ ​, 0 ≤ θ ≤ 2π 8. The vector v = ai + bj is perpendicular to the line ax + by = c. Use this fact to find an equation of the line through P perpendicular to v. Then draw a sketch of the line including v as a vector starting at the origin. ​P( − 1​, − 7​) v = − 4i + j The equation of the line is − 4x + y = − 3 . ​(Type an​equation.) Choose the correct graph of the line and vector below. A. B. 5 y C. 5 y 5 x -5 5 -5 D. y 5 x -5 5 -5 y x -5 5 -5 x -5 5 -5