Convincing Features
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Uploaded by Malaysia Assignment Help
Date
Submission Date: 5th October 2025 before 11:59:59pm.
Evidence of plagiarism or collusion will be taken seriously and the University regulations will be applied fully. You are advised to be familiar with the University’s definitions of plagiarism and collusion.
| (a) | Find the unit vector along the line joining point (2, 4, 4) to point (-3, 2, 2). | (4 marks) |
| (b) | Let A = 2ax + 5ay – 3az, B = 3ax – 4ay, and C = ax + ay + az. | |
| i. Determine A + 2B. | (2 marks) | |
| ii. Calculate . | (3 marks) | |
| iii. Find (A × B)/(A · B). | (6 marks) | |
| (c) | If A = 2ax + ay – 3az, B = ay – az, and C = 3ax + 5ay + 7az. | |
| i. A – 2B + C. | (2 marks) | |
| ii. C – 4(A + B). | (3 marks) |
| (a) | If the position of vectors of point T = 3ax – 2ay – az and point S = 4ax + 6ay + 2az, find: | |
| i. The coordinates of T and S. | (2 marks) | |
| ii. The distance vector from T to S. | (2 marks) | |
| iii. The distance between T and S. | (2 marks) | |
| (b) | If A = 5ax + 3ay – 2az, B = -ax + 4ay + 6az, and C = 8ax + 2ay, find the values of α and β such that αA + βB + C is parallel to the y-axis. | (8 marks) |
| (c) | Show that (A · B)2 + (A × B)2 = (AB)2. | (6 marks) |
| (a) | Calculate the angles that vector H = 3ax + 5ay – 8az makes with the x-, y-, and z-axes. | (6 marks) |
| (b) | Find the triple scalar product of P, Q and R, given that
P = 2ax – ay + az |
(6 marks) |
| (c) | Simplify the following expression. | |
| i. A × (A × B). | (4 marks) | |
| ii. A × [A × (A × B)]. | (4 marks) |
| (a) | Express the following points in Cartesian coordinates: | |
| i. P(1, 60°, 2). | (4 marks) | |
| ii. Q(2, 90°, -4). | (3 marks) | |
| iii. T(4,/2,/6). | (8 marks) | |
| (b) | Express the point P(1, -4, -3) in cylindrical and spherical coordinates. | (5 marks) |
| (a) | Transform the following vectors into cylindrical and spherical coordinates: | |
| i. D = (x + z)ay. | (8 marks) | |
| ii. E = (y2 – x2)ax +xyzay + (x2 – z2)az. | (12 marks) |
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