ENM1600 Engineering Mathematics, S1–2020 Assignment 1

Value: 10%. Due Date: Tuesday 24 March 2020.

• Submit your assignment electronically as one (1) PDF file via the link (Assignment 1 submission) available on study desk before the deadline. You may edit/change your assignment whilst in Draft status and it is before the deadline.

Please note: Once you have chosen “Submit assignment”, you will not be able to make any changes to your submission.

The submitted version will be marked so please check your assignment carefully before submitting.

• Hand-written work is more than welcome, provided you are neat and legible. Do not waste time type-setting and struggling with symbols. Rather show that you can use correct notation by hand and submit a scanned copy of your assignment. You may also type-set your answers if your software offers quality notation if you wish.

• Any requests for extensions should be made prior to the due date by contacting the examiner. Requests for an extension should be supported by documentary evidence.

An Assignment submitted after the deadline without an approved extension of time will be penalised. The penalty for late submission is a reduction by 5% of the maximum Assignment Mark, for each University Business Day or part day that the Assignment is late. An Assignment submitted more than ten University Business Days after the deadline will have a Mark of zero recorded for that Assignment.

• We expect a high standard of communication. Look at the worked examples in your texts, and the sample solutions in Tutorial Worked Examples to see the level you should aim at. Up to 15% of the marks may be deducted for poor language and notation.

• You may write a vector as either (a,b,c) or aˆi + bˆj + ckˆ, but be careful: you will lose marks if you mis-write these notations.

• Please note: No Assignments for Assessment purposes will be accepted after assignment solutions have been released.

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)

Value: 10%. Due Date: Tuesday 24 March 2020.

• Submit your assignment electronically as one (1) PDF file via the link (Assignment 1 submission) available on study desk before the deadline. You may edit/change your assignment whilst in Draft status and it is before the deadline.

Please note: Once you have chosen “Submit assignment”, you will not be able to make any changes to your submission.

The submitted version will be marked so please check your assignment carefully before submitting.

• Hand-written work is more than welcome, provided you are neat and legible. Do not waste time type-setting and struggling with symbols. Rather show that you can use correct notation by hand and submit a scanned copy of your assignment. You may also type-set your answers if your software offers quality notation if you wish.

• Any requests for extensions should be made prior to the due date by contacting the examiner. Requests for an extension should be supported by documentary evidence.

An Assignment submitted after the deadline without an approved extension of time will be penalised. The penalty for late submission is a reduction by 5% of the maximum Assignment Mark, for each University Business Day or part day that the Assignment is late. An Assignment submitted more than ten University Business Days after the deadline will have a Mark of zero recorded for that Assignment.

• We expect a high standard of communication. Look at the worked examples in your texts, and the sample solutions in Tutorial Worked Examples to see the level you should aim at. Up to 15% of the marks may be deducted for poor language and notation.

• You may write a vector as either (a,b,c) or aˆi + bˆj + ckˆ, but be careful: you will lose marks if you mis-write these notations.

• Please note: No Assignments for Assessment purposes will be accepted after assignment solutions have been released.

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)

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