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MCD4500 Engineering Mathematics

Assignment 1 (Trimester 1, 2024)

1. A linear transformation T : R3 → R3 satisfies T(1, 2, 4) = (1, 0, 0), T(3, 7, 14) = (0, 1, 0) and T(3, 4, 11) = (0, 0, 1). Determine the 3 × 3 matrix A such that T(v) = Av for all v ∈ R3.

[11 marks]

2. Consider the matrix

(a) Determine the two eigenvalues of A.

(b) i. Determine three linearly independent eigenvectors of A; and

ii. Verify that the eigenvectors found are indeed linearly independent.

[4 + (8 + 4) = 16 marks]

3. An underground pedestrian tunnel is being renovated as part of a city infrastructure project. A new staircase will be added, starting at some point P in the tunnel and travelling to the point Q on the surface. It is decided that the staircase should be as short as possible for the best public safety, convenience and cost. A construction company needs to know where to begin digging, how far, and in what direction. Their surveyor models the tunnel as an infinite straight line which passes through the points A = (28, 33, −8) and B = (−21, 5, −8), and the staircase as the infinite straight line the passes through the points P and Q = (15, 7, 0).

(a) Determine the coordinates of the point P.

(b) Determine the shortest distance d between between the tunnel and Q.

(c) Determine a unit vector v in the direction of the staircase.

[7 + 2 + 2 = 11 marks]

4. Define f : R 3 → R by f(x, y, z) = (xy2 − z) · e x−y+3z . Let P be the point (x, y, z) = (1, 1, 0), which lies on the surface S with the equation f(x, y, z) = 1.

(a) Determine the direction of the maximum rate of change of f at the point P.

(b) Determine the directional derivative of f at the point P in the direction of u = 8i + 4j − k.

(c) Determine the equation of the plane that is tangent to this surface S at the point P. Express the equation in the form ax + by + cz = d.

[5 + 2 + 2 = 9 marks]

Mathematical Communication: You will be assessed on your mathematical communication. Read the Student Writing Guide for assistance with this. You should attempt to implement good mathematical communication in all of your work. This includes:

❼ Clear and coherent sentences—including a concluding sentence—which should indicate the purpose of your working in the context of the question;

❼ Clear explanations and reasoning for key steps in your working; and

❼ Correct notation during calculations.

[3 marks]