Derive three simultaneous equations involving h d1 d2 and

Question

Q1)

a) if a = -4i + 2j + 5k, find the magnitude of a, and the unit vector parallel to a

b) if b = 6i + 5j -3k, find the angle a between a(part a) above) and b to 3 d.p.

c) if the points A, B and C (Fig 1.) have coordinates (12,-2,1), (7,6,-2) and (-3,-6,-1) respectively, find the area of triangle ABC

Fig 1.

Q2 The irregular pentagon, shown in Fig 2 below needs to be rotated through in an anti-clockwise direction about the origin of the system O

Fig 2.

Using Complex Numbers

i) convert each of z₁, z₂, z₃, z₄ and z₅ into exponential form
ii) rotate each point though anti-clockwise
iii) convert the results from ii) back into the cartesian form

Q3.

a) Using Cramer's Rule, or otherwise solve the system of equations for x₁, x₂, x₃

- x₁ - x₂ + 5x₃ = 6.5

- 2x₁ - 3x₂ + 4x₃ = -4.1

3x₁ + x₂ - 2x₃ = 3.3

b) By the method of LU decomposition solve the same system of equations

Q4

The cross-section of a cutting is shown. If AB = 76.218m, CD = 42.764m, the slope CB = 10:9, and the slope DA = 9:5

Fig 4.

i) Derive three simultaneous equations involving h, d1, d2, and put these three equations into matrix form AX = C.

ii) check det A ≠ 0,

iii) comment on why det A ≠ 0 is important

iv) and then, using Cramer's Rule, or otherwise, calculate h, d1, d2.

v) Calculate the area represented by ABCD in sq. metres to 3 decimal places.