2. LetA = 〈cos-, sin? and B = 〈cosi' sin, be two vectors on the x-y plane. Let -(cos-, sin π〉 3 4 be another non...
2. Let A = (cos, sin and B = (cos, sin) be two vectors on the x-y plane. Let C = (cos, sin be another non-zero vector on the x-y plane not collinear with A or B. Show that Ax B = -Bx C. If we could cancel B, as we could if these were real numbers, is it true that A= -Č? 2. Let A = (cos, sin and B = (cos, sin) be two vectors on the x-y...
Let Xn = a sin(bn+Z), where n ∈ Z, a, b ∈ [0, ∞) are constant, and Z has a continuous uniform distribution on [−π, π] (i.e. Z ∼ U([−π, π])). Show that Xn is stationary. (Hint: sin(x) sin(y) = 1 2 (cos(x − y) − cos(x + y)) may be helpful). l. Let Xn-a sin(bn+ Z), where n є z, a, b є lo,00) are constant, and Z has a continuous uniform distribution on [-π, π] (i.e. Z ~...
4. (a Let (sin( x cos( ) dr + (x cos(x + y) - 2) dy. dz= Show that dz is an exact differential and determine the corresponding function f(x,y) Hence solve the differential equation = z sin( Cos( y) 2 x cos( y) dy 10] (b) Find the solution of the differential equation d2y dy 2 y e dx dæ2 initial conditions th that satisfi 1 (0) [15] and y(0) 0 4. (a Let (sin( x cos( ) dr...
12. Let (o) and (1) represent the unit vectors along the x and y directions, respectively. The effects a rotation in the x-y plane. Show that the length of an operator(cosθ -sinθ) sin θ cos θ arbitrary vector (b) - a)+ b(), which is defined as Va2 + b", is unchanged by this rotation.
B 55 33 X Two vectors A and B are shown in the x-y plane. Vector A has a magnitude of 5 m and makes an angle of 55° with the positive x-axis vector B has a magnitude of 2 m and makes an angle of 33° with the negative x-axis. (See the figure above.) Vector C (not shown in the diagram) is the difference of A and B (C A - B) If you enter an expression, use degrees...
Let F(x, y, z) = x2y3 + y 2 sin(π z) /π + z2ex-1 a) Find the equation of the tangent plane to the graph of the function z = z(x, y) at the point (x, y) = (1, 1), if z satisfies the equation F(x, y, z) = 2 with z(1, 1) = 1. b) At the point P(1, 1, 1), determine in which of the two directions ~u = h−4, 3, 0i or ~v = h−3, 0, 4i...
9. Let V(x,y,+) - +w7+ sin(x + y)e'] + cos(x + 2) be a vector field in R'. Compute the curl and divergence of the vector field.
Let F(x, y, z) = sin yi + (x cos y + cos z)j – ysin zk be a vector field in R3. (a) Verify that F is a conservative vector field. (b) Find a potential function f such that F = Vf. (C) Use the fundamental theorem of line integrals to evaluate ScF. dr along the curve C: r(t) = sin ti + tj + 2tk, 0 < t < A/2.
7. Let A be a 4 x 3 matrix, and let b and y be two arbitrary vectors in R. We are told that the system Ax- b has a unique solution. What can you say about the number of solutions of the system Ax - y? Explain your answer. 8. Let u. v, w, b be arbitrary vectors in R". Suppose that b = x1u+xy+23w for some scalars i, r23. Show that Span u, v, w, b Span u,...
4. Let X have the following PDF: sin(x) , 0 < x < π , otherwise Ix(x) = 0 Find the CDF of X Using the Probability Integral Transformation Theorem, describe the process of generating values from the density of X Using R, generate 1,000 values using your process in part b. Produce a histogram of these generated values, and overlay the density curve of X over top. (Hint: in R, the function acos calculates the inverse cosine function.) Using...