Task 1:
Bob and Carol are attempting to solve a series of simultaneous equations using matrices to represent the equations, and provide a quick method of solution. Neither Bob nor Carol is familiar with this technique.
Your task is to help them both by demonstrating how matrices work.
(a) To demonstrate the basic principles of matrices, show Bob and Carol how to:
(i) add together the two matrices: ( 2 4 ) + ( -2 7 )
(ii) Multiply the following matrices:
(iii) Find the determinant of the following matrix
(iv) Find the inverse of the following matrix
(b) Bob has two simultaneous equations which need to be solved. By re-arranging them and forming a matrix equation, show him how you could use matrix algebra to solve them.
3x - 5y = -17.6
-2x + 7y - 22 = 0
(c) Carol has three simultaneous equations to solve and wants to use a technique called Gaussian Elimination" to do this. The three equations are:
3x + 2y - 3z = -7
x + 4y + z =25
2x + 6y + 3z =49
Show her how to use Gaussian elimination to solve these equations.
(d) For (b) and (c), use methods with which you are familiar to check your solutions to both parts. Further to this, critically reflect on the advantages and disadvantages of the techniques used to solve these problems. What are the main advantages and disadvantages to these analytical methods and what alternatives might you suggest?
Opportunities for Higher Grading
D1 In part (d) you need to critically reflect on your work, principally to your solutions to (b) and (c). A formal analysis is required here, suggesting alternative approaches with which you are familiar.
Task 2
Ted and represen Alice are working on some problems which involve representing systems of forces ted by vectors.
Your task is to help them solve the following problem:
(a) A force of (3i + 2j) Newtons acts at the point with position vector (2i - j) metres and a second force of (-3i + j) Newtons acts at the point with position vector (-3i - j) metres. Find the moment of the system about the point with position vector (I + 2j).
Alice is working on a new design of roof truss. She explains to Ted how the forces acting in the wooden sections could be resolved into vertical and horizontal components using trigonometry. Use this technique to answer the question below.
(b) Find the magnitude and direction of the resultant of the set of forces shown below:
