Show that the vector x = (1, 2) and the vector y = (3, 5) a base of form and determine all representations of z = (4, 7) related to the base
18. Consider the line L with vector equation (x, y, z)-(3, 4,-1 1,-2, 5) and the point P(2, 5, 7). Show that P is not on L, and then find a Cartesian equation for the plane that contains both P and L.
Solve in MATLAB Problem 3: Given the vector x- [2 1 0 6 2 3 5 3 4 1 3 2 10 12 4 2 7 9 2 4 51 use a for loop to (a) Add up the values of all elements in x (b) Compute the cumulative sum, y, of elements in x You can check your results using the built-in functions sum and cumsum. Q.5 What is the value of the sum of elements in vector x?...
6. Determine the gradbent vector test field of flx, y, z)=(x + y + 3 ) for the f(x, y, 21-30 2 = 0.25 7. Determine the volume of the solid that bies belo below x² + y + 3) and above Z= (x²+y") 1/2
1) Show that two lines are skew x+1 y+2 z+3 4:x=y=z and L: +7=5 2) Find the general equation of the plane containing the point P (1,2,3 ) and L, . 3) Find the point Q-the point of intersection the plane found in 2) and the line L. 4) Find the distance from the point (1,-1,2) to the line Lą.
3) Let X, Y be vector fields. For all functions f, define the commutator X, Y]0=X(Y()-Y(X(f). Show that X, Y=Z is a vector field, by verifying that it satisfies sum rule and product rule: Z(f+g)-Z(+Z(g) Zfg)-fZ(g)+gZ(). Extra credit: write /X, YJ in local coordinates.
2. Determine a vector that is perpendicular to both a=(5,3,-1) and 5=(4,- 2.7). Show that your newly created vector is perpendicular to both i and 7 4. Solve for x if ū=x,2 x,- 3 x and lūl=21.
x =-y+2 = -z+2 The symmetric equations for 2 lines in 3-D space are given as: 1. L,: x-2 = -y+1 = z+1 a) Show that lines L1 and L2 are skew lines. b) Find the distance between these 2 lines x =1-t y=-3+2t passes through the plane x+ y+z-4=0 2. The line Determine the position of the penetration point. a. Find the angle that the line forms with the plane normal vector n. This angle is also known as...
(1) Let G(,y, z) = (x,y, z). Show that there exists no vector field A : R3 -> R3 such that curl(A) Hint: compute its divergence G. (2) Let H R3 -> R3 be given as H(x,y, z) = (1,2,3). Find a vector potential A : R3 -> R3 such that curl(A) smooth function = H. Show that if A is a vector potential for H, then so is A+ Vf, for any f : R5 -> R (3) Let...
1. Consider the following system of linear equations: (8 marks) x+y = 3 7 7 2 -x+z=2 y-w=1 W = 4 z + w = 4 1) Use Gauss-Jordan elimination to put the augmented matrix corresponding to this system into reduced row echelon form. Clearly show all the elementary row operations applied. (3 marks) 12 nn
and Y ~ Geometric - 4 Let X ~ Geometric We assume that the random variables X and Y are statistically independent. Answer the following questions: a (3 marks) For all x E 10,1,2,...^, show that 2+1 P(X>x) P(x (3 = Similarly, for all y [0,1,2,...^, show that Show your working only for one of the two identities that are pre- sented above. Hint: You may use the following identity without proving it. For any non-negative integer (, we have:...