ENM1600 — Engineering Mathematics Assignment 1 Due: Tuesday 23 March 2021 Page 1 of 3ENM1600 — Engineering Mathematics Assignment 1 Due: Tuesday 23 March 2021 Page 1 of 3ENM1600 — Engineering Mathematics Assignment 1 Due: Tuesday 23 March 2021 Page 1 of 3
ENM1600 Engineering Mathematics, S1–2021
Assignment 1
Value: 10%. Due Date: Tuesday 23 March 2021.
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ENM1600 — Engineering Mathematics Assignment 1 Due: Tuesday 23 March 2021 Page 2 of 3ENM1600 — Engineering Mathematics Assignment 1 Due: Tuesday 23 March 2021 Page 2 of 3ENM1600 — Engineering Mathematics Assignment 1 Due: Tuesday 23 March 2021 Page 2 of 3
QUESTION 1 (24 marks)
Four forces (measured in Newtons) act on a crate. The forces are given by the vectors
F1 = 379 î− 104 ĵ + 21 k̂ N, F2 = −165 î + 66 ĵ + 110 k̂ N,
F3 = 72 î− 144 ĵ + 99 k̂ N, and F4 = −321 k̂ 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 (measured in metres) is given by
d = 5.5 î− 9 ĵ + 0.5 k̂m
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 (1.2, 7.6,−1.6) m, what is the moment, M = r× F3, at
the point (3.6, 2.8,−1.6) m? (6 marks)
(d) What is the angle (in degrees to one decimal place) between the two force vectors F2 and F3?
(8 marks)
QUESTION 2 (16 marks)
(a) An aircraft, flying in a straight line, passes through the points (−12, 5, 4) and (−2, 7, 3) measured
in kilometres (km). 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 5x− 3y + 40z = 13 where x, y, and z are measured in kilometres (km). (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, 3) (in metres). Find the vectors
−→
AB and
−→
AC and hence the equation of the
plane through these three points. (10 marks)
0
x
z
y
3 m
A
3 m
2 m
3 m
C
5 m
2 m
2 m
B
4 m
3 m
ENM1600 — Engineering Mathematics Assignment 1 Due: Tuesday 23 March 2021 Page 3 of 3ENM1600 — Engineering Mathematics Assignment 1 Due: Tuesday 23 March 2021 Page 3 of 3ENM1600 — Engineering Mathematics Assignment 1 Due: Tuesday 23 March 2021 Page 3 of 3
QUESTION 3 (20 marks)
Find the following given the matrices (show your working)
A =
−1 9 0
2 −5 1
3 0 −8
2 −7 6
, B =
7 0 −4 5−4 3 9 2
3 −5 0 −2
, and C =
−1 6 21 −2 0
5 −3 y
(a) Evaluate BT − 7A; (8 marks)
(b) Use matrix multiplication to find BA. Show your working; (8 marks)
(c) Find detC in terms of y by expanding along any column.
For what value of y will the matrix C be singular? Show your working. (4 marks)
QUESTION 4 (8 marks)
Suppose the matrix product AB is defined.
(a) If B is a 42× 81 matrix and A is a row matrix, give the dimensions of A and AB. (4 marks)
(b) Suppose A + AT is defined and AB is a 325 × 121 matrix. What is the size of matrix A?
(2 marks)
(c) If A is a 91× 101 matrix and (AB)T is a 37× 91 matrix, what size is matrix B? (2 marks)
QUESTION 5 (12 marks)
Find the determinant of the matrices below (where x is a scalar) by inspection. That is by using the
properties of determinants and not by direct evaluation. Give your reason(s) in each case.
A =
−4 0 0 0
8 x 0 0
1 −5 7 0
−3 x −2 x
B =
XXXXXXXXXX −15
−4x −22 2x 11
28 26 −14 −13
35 − XXXXXXXXXX
C =
8 0 0 0
−3 x −2 x
1 −5 7 0
8 x 0 0
Hint: For matrix C compare with matrix A.
QUESTION 6 (20 marks)
Given matrix B answer the following.
B =
7 −1 36 −5 0
−9 2 8
(a) Find the adjoint matrix for the matrix B. Show your working; (15 marks)
(b) Determine if the matrix B 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 B−1B. (5 marks)
End of Assignment XXXXXXXXXXmarks Total)