Assignment Title - Vector Geometry And Matrix Methods
Learning Objective : Be able to analyse and model engineering situations and solve problems using vector geometry and matrix methods
1 Represent force systems, motion parameters and waveforms as vectors and determine required engineering parameters using analytical and graphical methods.
2 Represent linear vector equations in matrix form and solve the system of linear equations using Gaussian elimination.
3 Use vector geometry to model and solve appropriate engineering problems.
Purpose of this assignment is to: provide the analytical knowledge and techniques needed to carry out a range of engineering tasks and will provide a base for further study of engineering mathematics.
Scenario :
Now you are working in one of the biggest electrical distributed factory, as a junior mathematical analyst. If some problem will happen, you have to find out the errors and solve the problem to reach a normal position. And you have been asked by your line manager to find the solutions to some of the problems.
LO3: P3.1: Represent force systems, motion parameters and waveforms as vector and determine required engineering parameters using analytical and graphical methods.
Task 1
(i) Find the resultant of the following forces : (a) p = iˆ + 3 jˆ - kˆ,q= 5iˆ - 2kˆ,u= - jˆ + 3kˆ.
(ii) Find the work done by a force p acting on a particle, if the particle is displaced from a point A to a point B alone a straight segments AB where, A: (4,1,3),B: (5,3.5),P= iˆ + jˆ - kˆ.
(iii) Write a vector equation of the line that passes through the point P and is parallel to v. Then write parametric equations of the line.
(a) P(3, -5) , v=(4,2)
(b) P(-1, 9) , v=(4,2)
(c) Use the parametric equations in (b) to write the equation of the line in slope intercept form.
(iv) A particle is moving along the curve having parametric equations: x = 4 cos 1t and y = 4 sin 1t.
2 If x and y are centimetre measures, find the speed and the magnitude of the particles acceleration vector at t seconds.
Draw a graph of the particles path, and also draw the representations of the velocity and acceleration vectors having initial points where t= 1p.
2: Represent linear vector equations in matrix form and solve the system of linear equations using Gaussian elimination.
Task (2)
(2) Convert the given linear vector equations in matrix form and sole the system of linear equations by using Gaussian Elimination Method:
(a) xv1 + yv2 + zv3 = b ;
where v1 =
é1ù ê2ú
, v2 =
é2 ù ê-1ú, v3 =
é3ù ê1úand b =
é9ù ê8úare vectors.
ê ú ê ú ê ú ê ú
êë3úû
êë-1úû
êë0úû
êë3úû
(b) xu1 + yu2 + zu3 = d ;
where u1 =
é1 ù
ê-1ú
, u2 =
é2ù ê3ú, u3 =
é3ù ê2úand d =
é0ù ê0úare vectors.
ê ú ê ú ê ú ê ú
For D1
êë2 úû
êë1úû
êë2úû
êë0úû
Check your results in equation (b), by your own consideration.
LO3: P 3.3: Use vector geometry to model and solve appropriate engineering problems.
Task (3)
(3) (i) A current phasor i1 is 6A and horizontal. A second phasor i2 is 9A and is at 60° to the horizontal. Determine the resultant of the two phasors , i1 + i2 , and the angle the resultant makes with current i1, by
(a) the ‘nose – to – tail’ method, and
For M2
(b) the parallelogram method.
(ii) A steady fluid motion has velocity v =
xˆi + yˆi. Find curl v.
How would you know the motion is compressible or incompressible?