Show that the Cauchy-Riemann equations [see (2.2) and Problem 2.46] in a general orthogonal curvilinear
where, as in Chapter 10, the variables are
Hint: Consider the directional derivatives (Chapter 6, Section 6 ) in two perpendicular directions. (Compare Problem 2.46.) Also show that
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- 3. Give the parametric and symmetric equations of the line that passes through the point (4, –3,5) and is perpendicular to both vectors (2,6, 1) and (1, 3, –5).arrow_forward2. Write the solution set of Ax = b in parametric vector form. Hint: Make sure to compute the reduced echelon form of A before writing the solution set. If you use the echelon form given, you’ll have basic variables in your answer which is not okay.arrow_forwardThe question is in the screenshotarrow_forward
- 3. For both proofs in this problem, you may not write or refer to any of the entries of any vector. (Hint: Use only the properties from §2.1 Theorem 1 from the lecture notes.) (a) Let a € Rn be a vector satisfying b + a = b for all b € R". Prove that a = 0. (b) Let x = R", and let y € R" be such that x +y = 0. Prove that y = -x.arrow_forwardThe vector F = −a + 2a expressed in RCS and in terms of ø isarrow_forwardWeek 12: Part 1: This exercise will ask you to explore the equality Null(A") = (Col(A))-. Pick a vector u in Rª and a vector v such that u and v are orthogonal; then find a vector w which cience and Technology Unleashed Interview Dec 7, 202U 1he 7 is orthogonal to both u and v. Finally, find a vector z which is orthogonal to u, v, w. For the last two steps, you should realize that you are solving a system of linear equations, and if you write them down that the system is represented by AT where the columns of A are the vectors you are trying to be orthogonal to. Now, verify with computations that the set {u, v, w, z} is linearly independent. If any of the vectors u, v, w, z are scalars of the standard basis vectors e1, e2, €3, e4 then start over. Set the matrix P = [u_v w z] and compute without calculations the vectors P-'u, P-'v, P-lw, and P-lz.arrow_forward
- Chapter 11, Section 11.3, Question 033 If L is a line in 2-space or 3-space that passes through the points A and B, then the distance from point P to the line L is equal to the length of the component of the vector AP that is orthogonal to the vector AB. Use the method above to find the distance from the point P(-3, 1,7) to the line through A(1, 1.0) and B(-2,3, -4). Distance = Edit Click if you would like to Show Work for this question: Open Show Workarrow_forward1. The distance of a point in the 3-D system from the origin a. is defined by the absolute value of the vector from the origin to this point. b. is the square root of the square of the sums of the x-, y- and z-values. c. is the square root of the sum of the squares of x-, y- and z-values. d. can either be negative or positive. e. None of the above. 2. In parametrizing lines connected by two points in 3-D plane, a. there is only one correct parametrization. b. symmetry equations may not exist. c. a, b, and c must not be equal to 0. d. the vector that connects the two points is a scalar multiple of the vector containing the direction numbers. e. None of the above. 3. A plane in 3D-space system a. is generated by at least three points. b. can lie in more than one octant. c. must have a z-dimension. d. must have a point other than the origin. e. None of the above. 4. A quadric surface a. must have either x2, y2, or z2 or a combination of those, on its general expression. b. must have a…arrow_forward3. a. b. i. You are given the vector form of the line [x, y] = [3, 1] +t[-2, 5] and a Point P. When you write the parametric equations of the line, what should you notice about the value of t? Check each of the following points to see if they lie line, (-1, 11) ii. (9, -15) (Hint: Find the value of t for both x an y, look for a consistent solution).arrow_forward
- 3 and w = |4 2 are linearly independent. 1. (a) Show that the vectors v = -3 (b) Explain why 3 vectors in R" of the form u, v and w = au + bv, where a, any real numbers, are linearly dependent. arearrow_forwardPART 4 1 YYYUTY 17. find the equation of the line through point (-1,2,3) and perpendicular to the line for which cos α = ²/3, cox B= - 1²/ a cos y = ²/ 2 3 18. Find the equation of the plane through the point (-1,2,4) and parallel to the plane 2x-3y-52+6=0 2 ( x + 1) + (− 3) ( y − 2) + (~^ s) ( ² − 4 ) = 0 1x - 31 -57 +34=0arrow_forwardSolution. Let T: R" → R™ be a linear transformation and let {V₁,..., vp} be a set of linearly independent vectors in R". Suppose that {T(v₁),...,T(v₂)} is inearly dependent. Then, there are real scalars c₁,..., Cp, not all zero, such that C₁T (V₁) + + cpT(vp) = 0.arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageLinear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning