ENM1600 Engineering Mathematics Assignment 1
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• 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.
Four forces (measured in Newtons) act on a crate. The forces are given by the vectors
F1 = 374ˆi − 61ˆj + 82 kˆ N, F2 = −234ˆi + 13ˆj + 78 kˆ N,
F3 = 36ˆi − 216ˆj + 168 kˆ N, and F4 = −295 kˆ N.
(a) What is the resultant, R = F1 + F2 + F3 + F4, of the four forces? What is the magnitude of the resultant force?
(b) If the displacement of the crate is given by d = 3.5ˆi − 4.5ˆj + 0.25 kˆ m
what is the contribution by the first force F1 to the work done i.e. W = F1 · d?
(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?
(d) What is the angle (in degrees to 4 significant figures) between the force vectors F2 and F3?
(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.
(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.
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.
(b) Suppose A is an identity matrix and AB is a 84 × 63 matrix. What is the size of matrix A?
(c) If B is a 74 × 63 matrix and (AB) T is a 63 × 95 matrix, what size is matrix A?
(a) Find the adjoint matrix for the matrix R. Show your working;
(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.
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