Recent Question/Assignment

ENM1600 Engineering Mathematics, S1–2020 Assignment 1
Value: 10%. Due Date: Tuesday 24 March 2020.
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QUESTION 1 (24 marks)

Four forces (measured in Newtons) act on a crate. The forces are given by the vectors
F1 = 374ˆi- 61ˆj + 82kˆ N, F2 = -234ˆi + 13ˆj + 78kˆ N, F3 = 36ˆi- 216ˆj + 168kˆ N, and F4 = -295kˆ N.
(a) What is the resultant, R = F1 + F2 + F3 + F4, of the four forces? What is the magnitude of the resultant force? (6 marks)
(b) If the displacement of the crate is given by
d = 3.5ˆi- 4.5ˆj + 0.25kˆm
what is the contribution by the first force F1 to the work done i.e. W = F1 ·d? (4 marks)
(c) If the third force, F3, acts at the point (0.9,-0.1,2.1)m, what is the moment, M = r × F3, at the point (0.9,5.3,-2.1)m? (6 marks)
(d) What is the angle (in degrees to 4 significant figures) between the force vectors F2 and F3?
(8 marks)
QUESTION 2 (16 marks)

(a) An aircraft, flying in a straight line, passes through the points (11,-3,2) and (9,9,3) measured in kilometres. Find the equation of the line that describes the flight path of the aircraft. Determine whether or not the aircraft is flying parallel to the ground which is described by the plane
6x - 5y + 72z = 42. (6 marks)
(b) A large shade sail is to be hung tautly between the tops of 3 poles. The tops of these poles are
located at points A, B, and C as shown in the figure below. For instance the point A is located
-? -?
at (-2,-3,2) (in metres). Find the vectors AB and AC and hence the equation of the plane through these three points. (10 marks)
C

QUESTION 3 (20 marks)

Find the following given the matrices
? ?
2 4 2
A, and C = ? 1 0 -1 ?
? ?
6 x -1
(a) Evaluate 3A-BT; (8 marks)
(b) Use matrix multiplication to find BA. Show your working; (8 marks)
(c) Find detC in terms of x by expanding along any column.
For what value of x will the matrix C be singular? Show your working. (4 marks)
QUESTION 4 (8 marks)

Suppose the matrix product AB is defined.
(a) If A is a 42×19 matrix and B is a column matrix, give the dimensions of B and AB. (4 marks)
(b) Suppose A is an identity matrix and AB is a 84 × 63 matrix. What is the size of matrix A?
(2 marks)
(c) If B is a 74 × 63 matrix and (AB)T is a 63 × 95 matrix, what size is matrix A? (2 marks)
QUESTION 5 (12 marks)

Find the determinant of the matrices below by inspection. That is by using the properties of determinants and not by direct evaluation (note: C = AAT). Give your reason(s) in each case.
? ? ?
-3 2 -5 1 5 -60
? 0 7 1 -6 ?? ?? -4x 48
?
A = ? ? B = ?
? 0 0 -2 8 ? ? -1 12
? ? ?
0 0 0 x 2x -24
QUESTION 6 (20 marks)

Given matrix R answer the following. 25
20 -5
10 -15 ?
12 ??
? 3 ?
?
6 ?
? C = ??
?
? 39
3
18 x 3
86 -50
-6x 18
-50
68
8x ?
x
-6x ??
? 8x ?
?
x2
?
5
R = ? 1
?
-2 -1
0
-4 4 ?
-7 ?
?
3
(a) Find the adjoint matrix for the matrix R. Show your working; (15 marks)
(b) Determine if the matrix R is invertible and, if possible, find its inverse using the results from part (a). Show your working.
Confirm you have found the inverse (if it exists) by calculating R-1R. (5 marks)

End of Assignment 1 (100 marks Total)