Fnd the perpendicular distance of the plane 5 x 2 y - z

Question

Question #1. Find the perpendicular distance of the plane 5 x + 2 y - z = -22 from origin 0 by first finding the co-ordinates of the point P on the plane such that OP is perpendicular to the given plane.

Question #2. We have an empty cardboard box of dimensions breadth = 4 az length = 5 in, and height = 6 m. An ant and a fly are sitting at one of the 8 comers inside the box and wish to go get some food at the corner diagonally across (the farthest point in the box for them). What is the minimum distance the ant must travel before it gets to the food? How about the fly?

Question #3. Let N = 3 and M = 2N. M-th root of 1 is given by ω =-α, where α =exg[2Πj/N] j = √-1. It is verified as

 ω^M = ω^2N= (-α)2N = (-1)^2N exp[-4Πj] = cos4Π -j sin4Π = 1.

Note that (-1)² = 1. Using this M-th root of 1, we create an (M xM) square matrix R. The (a. b)-th element of R is given by

r_ab= ω^(a-1)(b-1 a = 1,2,........m b = 1,2..........m 

A. Write the matrix R expressing its elements in terms of α. Show at least the top 3 x 3 part and all the elements on the four corners. Is R a Hermitian matrix?

B. Consider another (M x M) square matrix S such that the (c, d)-th element of S is given by

s_c,d =ω^(c-1)(d-1) c =1, 2, ...,M; d =1,Z ...,M

Let T = SR Write the elements of T. Compute determinant of T.

C It is claimed that inverse of S, S^-1 = (1/M)R It Is that correct? Express R^-1 in terms of S. Justify.

D. Find the rank of the matrix S and its nullity. Are all the rows of S Li?

E. Solve the linear system Rx = b for M= 6 and b = [1 1 1 1 1 l]^t.

You may find these very useful:

exp[ja] = cosα +j sinα;                     1+α+α² ...+α^M-1= [1-α^M]/(1-α).