2.10. Orthogonally diagonalize the rotation matrix -sin α cos α sin α cos α where α is not a mult...
Where is the singularity in the following 3-sequential Euler Angle Rotation matrix about the ZY'Z'' axis where Y' is the new Y axis after the first Z rotation and Z'' is the new Z axis after the Y' rotation. The matrix has been generated seen below, but I'm having trouble finding the singularity: C1S2 S1S2 , subscrip t 1 = e, therefore c1 is cos(6), cosine of the first rotation angle value Note: c is Cos, s
In the 3D Cartesian system the rotation matrix is around the z-axis is (a 2D rotation): Where is the angle to rotate. Then rotation from A to A' is can be represented via matrix multiplications: [A'] = [R][A] Such a rotation is useful to return a system solved in simplified co-ordinates to it's original co-ordinate system, returning to original meaning to the answer. A full 3D rotation is simply a series of 2D rotations (with the appropriate matrices) Question: If...
4. (15 marks) Consider the following equation: where i denotes the complex number satisfying i2--1 (a) Rewrite the number -i in the exponential form and transform equation (5) into (b) Solve (6) to get the five solutions wo, ..., wa and draw them on the Argand diagramme (c) Show that wo··· , ua are the eigenvalues of the following real-valued matrix 0 0 0 0 0 cos(2m/5) A-10 -sin2(2π/5) 0 0 cos(2π/5) 0 0 0 2cos(4π/5) 2 Hint: compute the...
Suppose firm j’s output is given by yj = n 1−α j , where 0 < α < 1 (α is a parameter). Suppose the firm must pay a fixed cost b < α if it wants to operate. That is, the firm’s profits are given by π (nj ) = 0 , if nj = 0 and π (nj ) = n 1−α j − wnj − b , if nj > 0 where w is the wage. (a)...
Problem 2 Show that for an index ofrefraction k that D(a) is minimized when sin α- 4 Use this to obtain p- arcsin to obtain ß arcsinand fil in the following table. 4-k2 3k2 a in degrees B in degrees R(a) in degrees Color Red Orange Yellow Green Bluc Indigo Violet 1.331 1.332 1.333 1.335 1337 1.340 1.344 The values of R(a) you obtained in the table accounts for the various color bands that appear in the rainbow as sunlight...
(1 point) Let Vf =-8xe-r sin(5y) 20e-x. cos(Sy) j. Find the change inf between (0,0) and (1, π/2) in two ways vf . dr, where C is a curve connecting (0,0) and (1.d2). (a) First, find the change by computing the line integral The simplest curve is the line segment joining these points. Parameterize it: with 03t s 1, r(t)- so that Icvf . di- Note that this isn't a very pleasant integral to evaluate by hand (though we could...
→ (1 point) Let Vf-6xe-r sin(5y) +1 5e* cos(Sy) j. Find the change inf between (0,0) and (1, n/2) in two ways. (a) First, find the change by computing the line integral c Vf di, where C is a curve connecting (0,0) and (1, π/2) The simplest curve is the line segment joining these points. Parameterize it: with 0 t 1, K) = dt Note that this isn't a very pleasant integral to evaluate by hand (though we could easily...
Problem 1: Consider a 2nd order homogeneous differential equation of the form aa2y"(x)bay(x) + cy = 0 (1) where a, b, c are constants satisfy so that y(x) = x (a) Find and justify what conditions should a constant m to (1) is a solution (b) Using your solution to (1) Write these three different cases as an equation that a, b,c satisfy. Hint: Use the quadratic formula we should get three different cases for the values that m can...
1 Cylindrical coordinate system Given the relation of the cylindrical coordinate system r=r cos pi+r sin øj + zk (1) Lets define vectors er, eg, and ez, that indicate the direction of the vectors in the cylindrical coordinate system. Using the definition ar e = pt=r, p2 = 0, p3 (2) (a) Find a matrix for calculating er, er and e, in terms of i, j, and k. Invert the relation for expressing i, j, and k in terms of...
Consider a cylindrical capacitor like that shown in Fig. 24.6. Let d = rb − ra be the spacing between the inner and outer conductors. (a) Let the radii of the two conductors be only slightly different, so that d << ra. Show that the result derived in Example 24.4 (Section 24.1) for the capacitance of a cylindrical capacitor then reduces to Eq. (24.2), the equation for the capacitance of a parallel-plate capacitor, with A being the surface area of...