Question #1. Find perpendicular distance of the plane 6x + y + 2z = 18 from origin O by first finding 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 = 3 m, length = 4 m, and height = 2 m. An ant and a fly are sitting at one of the 8 corners 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 fly must travel before it gets to the food? How about the ant?
Question #3. Let N = 3 and M = 2N. M-th root of 1 is given by ω = -α, where α = exp[- 2Πj/N], j = √-1. Note j2 = -1. It is verified as
ωM = ω2N = (-α)2N = (-1)2N exp[-4Πj] = cos 4Π - j sin 4Π = 1.
Using this M-th root of 1, create an (M x M) square matrix R. The (a, b)-th element of R is given by
ra,b = 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
sc,d = ω-(c-1)(d -1), c = 1, 2, ..., M; d = 1, 2, ..., M.
Let T = SR. Write elements of T. Compute determinant of T.
C. Express R-1 in terms of S. Show your work.
D. Find the rank of the matrix R and its nullity. Are all the rows of S l.i.?
E. Solve the linear system Sx = b for M = 6 and b = [1 1 1 1 1 1]t. You may define x = Ry and solve for y first.
You may find these very useful:
exp[ja] = cos a + j sin a; 1 + a + a2 + ... + aM - 1 = [1 - aM]/(1 - a).