Sect.12.4 Enhanced Assignment

Instructor: Shijun Zheng Course: Spring 2023 - Calculus III (MATH2243-A-B) Student: Jalen Wilson Date: 07/11/24 Assignment: Sect.12.4 Enhanced Assign 1. Find the length and direction​(when defined) of u × v and v × u. u = 4i − 2j − 3k​, v = 3i − 3k The length of u × v is 9 . ​(Type an exact​answer, using radicals as needed.) Select the correct choice below​and, if​necessary, fill in the answer boxes to complete your choice. A. The direction of u × v is 2 i+ 3 ​(Type exact​answers, using radicals as​needed.) 1 j+ 3 2 k. 3 B. The direction of u × v is undefined. The length of v × u is 9 . ​(Type an exact​answer, using radicals as needed.) Select the correct choice below​and, if​necessary, fill in the answer boxes to complete your choice. A. The direction of v × u is 2 i+ 3 ​(Type exact​answers, using radicals as​needed.) − − 1 j+ 3 − 2 3 k. B. The direction of v × u is undefined. 2. Find the length and direction​(when defined) of u × v and v × u. u = − 3i + 9j − 9k​, u×v = 0 v = − i + 3j − 3k ​(Simplify your​answer.) Select the correct choice and fill in any answer boxes in your choice below. A. The direction of u × v is i+ j+ k. ​(Simplify your​answers, including any radicals. Use integers or fractions for any numbers in the​expressions.) B. The direction of u × v is not defined. v×u = 0 ​(Simplify your​answer.) Select the correct choice and fill in any answer boxes in your choice below. A. The direction of v × u is i+ j+ k. (​ Simplify your​answers, including any radicals. Use integers or fractions for any numbers in the​expressions.) B. The direction of v × u is not defined. 3. Sketch the coordinate axes and then include the vectors u​, v​, and u × v as vectors starting from the origin. u = − 6i​, v = − 8j Which of the following graphs contains u​, v​, and u × v​? +z A. +z B. u×v v u +x C. D. +z u×v +y u +x +y +y v u×v +z v u +x u v u×v +y +x 4. Sketch the coordinate axes and then include the vectors u​, v​, and u × v as vectors starting at the origin. u=i−k v=j+k Choose the correct answer below. 1 z A. z B. u v y u×v x u x z C. u × v y u v y v z D. v x u×v u v y x u×v 5. Give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations. 2 2 2 x + y + (z + 7) = 100​, z = 1 Choose the correct description. A. The line through (6,6,1) parallel to the​z-axis B. The line through ​(6,0,1​) and ​(0,6​,1​) C. The circle with center ​(0,0,1​) and radius 6​, parallel to the​xy-plane D. The circle with center ​(0,0,1​) and radius 36​, parallel to the​xy-plane 6. Let u = 7i − j + k​, v = − 14i + 2j − 2k​, w = j − 7k. Which​vectors, if any are ​(a)​perpendicular? ​(b) ​Parallel? ​(a) Which vectors are​perpendicular? Select the correct choice below and fill in the answer​box(es) within your choice. A. The vectors as​needed.) are perpendicular. ​(Use a comma to separate answers B. Vector is perpendicular to vectors separate answers as​needed.) . ​(Use a comma to C. None of the vectors are perpendicular. ​(b) Which vectors are​parallel? A. The parallel vectors are B. u,v . ​(Use a comma to separate answers as​needed.) is one pair of parallel vectors. Another pair of parallel vectors is . ​(Use a comma to separate answers as​needed.) C. None of the vectors are parallel. 7. Determine whether the following statements are always​true, or not always true. Give reasons with the answers. Answer parts a through h. a. Is u = u • u always true or not always​true? Explain the answer. A. This is not true for the zero vector. B. This is always true because u = 2 2 u1 + u2 + ... = u1 u2 + u2 u1 + ... = u•u. C. This is not true when components of the vector are negative. D. This is always true because u = 2 2 u1 + u2 + ... = u1 u1 + u2 u2 + ... = u•u. b. Is u • u = u always true or not always​true? Explain the answer. A. This is not always true because u • u = 2 2 u1 + u2 + ... ​, while u = u1 u1 + u2 u2 + .... Therefore only certain vectors make the statement true. B. This is always true because this is the definition of a cross product. C. This is not always true because u • u = u1 u1 + u2 u2 + ...​, while u = 2 2 u1 + u2 + ... . Therefore only certain vectors make the statement true. D. This is always true because this is a property of the cross product. c. Is u × 0 = 0 × u = 0 always true or not always​true? Explain the answer. A. This is always true because u × 0 = 0 and 0 × u = 0. Hence both cross products result in the scalar 0. B. This is not true when u is the zero vector as well. C. This is always true because u × 0 = 0 and 0 × u = 0. ​Hence, u × 0 = 0 × u = 0. D. This is not true when some of the components of the vector are zero. d. Is u × ( − u) = 0 always true or not always​true? Explain the answer. A. This is always true because when calculating the cross product as a​determinant, the determinant of the 2 by 2 matrices for each term is 0. B. This is always true because u × ( − u) suggests addition and the zero vector follows. C. This is not always true because the product of some vectors and​it's opposite is negative. D. This is not always true because when calculating the cross product as a​determinant, the determinant of the 2 by 2 matrices for each term is 1. e. Is u × v = v × u always true or not always​true? Explain the answer. A. This is not always true because by​definition, v × u = − ​(u × v​). B. This is not always true because by​definition, v × u = ​(u • v​). C. This is always true by the commutative property of multiplication. D. This is always true because it is a property of a cross product. f. Is u × (v + w) = u × v + u × w always true or not always​true? Explain the answer. A. This is not true when vectors v and w are parallel. B. This is always true by the distributive property. C. This is not always true because when the vectors v and w go in opposite directions this is not true. D. This is always true because it is a property of a cross product. g. Is (u × v) • v = 0 always true or not always​true? Explain the answer. A. This is always true because by​definition, the cross product u × v is perpendicular to both vectors u and v. B. This is always true because by​definition, the cross product u × v is perpendicular to only vectors v. C. This is not always true because when v has negative components the vector is postive. D. This is not always true because when v has negative components the scalar is postive. h. Is (u × v) • w = u • (v × w) always true or not always​true? Explain the answer. A. This is not always true when one of the vectors is the zero vector. B. This is always true by the associative property of multiplication. C. This is always true because after expanding (u × v) • w​, the terms can be rearranged and​u-term may be factored from each pair of terms. This new expression can be rewritten as u • (v × w). D. This is not always true when two of the vectors are parallel. 8. Given nonzero vectors u​, v​, and w​, use dot product and cross product​notation, as​appropriate, to describe the following. Answer parts a through f. a. Describe the​statement, "The vector projection of u onto v,​" using dot product and cross product notation. u•v v A. proju v = v•v B. projv u = C. proju v = u•v v•v v•u u•u v u v•u u D. projv u = u•u b. Describe the statement, "A vector orthogonal to u and v​," using dot product and cross product notation. A. ± (u + v) • u B. ± u • v C. ± (u − v) × v D. ± u × v c. Describe the​statement, "A vector orthogonal to u × v and w​," using dot product and cross product notation. A. ± (u × v) × w B. ± (u × v) • w C. ± u × v × w D. ± w • (u × v) d. Describe the statement, "The volume of parallelepiped determined by u​, v​, and w​," using dot product and cross product notation. A. (u • v) × w B. (u × v) • w C. (v × u) • w D. (u × v) • w e. Describe the​statement, "A vector orthogonal to u × v and u × w​," using dot product and cross product notation. A. (u × v) • (u × w) B. v × w C. (u × v) × w D. (u × v) × (u × w) f. Describe the​statement, "A vector of length u in the direction of v​," using dot product and cross product notation. A. uv B. u C. u D. v v v v v u u