MTD203: Given a = (6, -2, 9) and b = (5, 6, -8), solve the Following vector Arithmetic: Advanced Graphics Design Assignment, SUSS, Singapore

Question 1
(a) Given a = (6, -2, 9) and b = (5, 6, -8), solve the following vector arithmetic:
(i) 3a – 9b
(ii) 7a + 4b
(iii) |7a + 4b|

(b) Consider the triangle whose vertices lie in the point P0(-7, 2, 5), P1(3, 6, -4) and P2(9, -1, 8).
(i) Using the cross product, calculate the area of the triangle.
(ii) Using dot product, calculate each of the three angles of the triangle.

Question 2
(a) Describe the operations of multiplying two matrices by performing the following tasks. Given two matrices

(i) Calculate AB, showing the intermediate steps.
(ii) Calculate the determinants of both A and B, showing the intermediate steps.
(iii) Calculate the determinant of AB, showing the intermediate steps. Verify that
det(A) x det(B) = det(AB) .

(b) Solve the inverse of the following matrix, showing the intermediate steps

Question 3
A given surface is defined by the equation = 1. A ray starts at the
source S(-3, -2, -4) and moves in the direction V(2, 1, 3). Describe the intersection of the ray and the surface as follows:
(a) Identify and describe the geometrical shape that is defined by the equation of the surface above? Illustrate the shape with a sketch or diagram.
(b) Calculate the intersection between the path of the ray and the surface.
(c) Using your answer in part (b), determine the point where the ray hits the surface at the first instance.

Question 4
(a) A light ray hits a perfectly shiny surface with L= (5, 4, 8) and N= (4, 8, 5).
(i) Normalise L and N.
(ii) Calculate the reflection vector in this instance.

(b) Solve the transformation operation that involves rotating a point P(7, 5, 2) by 30 anticlockwise about the x-axis, next, 30 anticlockwise about the y-axis, and finally 30 anticlockwise about the z-axis. Compute the final position of the point.