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MAT1503 Assignment 5 2023
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MAT1503ASSIGNMENT 5 2023
, MAT1503/101/0/2023
Tutorial Letter 101/0/2023
LINEAR ALGEBRA I
MAT1503
Year Module
Department of Mathematical Sciences
This tutorial letter contains important information about Assignment 5.
BARCODE
, ASSIGNMENT 05
Due date: Wednesday, 2 August 2023
Total Marks: 100
ONLY FOR YEAR MODULE
This assignment cover chapter 3 of the prescribed book as well as the study guide,
it is specifically based on All chapters of the Study Guide & Appendix B and Chapter 3
of HC
Question 1: 12 Marks
(1.1) Let U and V be the planes given by: (2)
U : λ x + 5y − 2λ z − 3 = 0,
V : −λ x + y + 2z + 1 = 0.
Determine for which value(s) of λ the planes U and V are:
(a) orthogonal, (2)
(b) Parallel. (2)
(1.2) Find an equation for the plane that passes through the origin (0, 0, 0)and is parallel to the (3)
plane − x + 3y − 2z = 6.
(1.3) Find the distance between the point (− 1, − 2, 0) and the plane 3x − y + 4z = − 2. (3)
Question 2: 9 Marks
(2.1) Find the angle between the two vectors ⃗v = ⟨− 1, 1, 0,− 1⟩ ⃗v = ⟨ 1, − 1, 3, − 2⟩ . Determine (3)
whether both vectors are perpendicular, parallel or neither.
(2.2) Find the direction cosines and the direction angles for the vector ⃗r = ⟨ 0, − 1, − 2, 34 ⟩ . (3)
(2.3) HMW:Additional Exercises.
Let ⃗r (t ) = ⟨ t ,− 1t , t2 − 2⟩ . Evaluate the derivative of ⃗r (t |)t=1 . Calculate the derivative of
V (t )· ⃗r (t )whenever V (1) = ⟨− 1, 1, − 3⟩ and V ′ (1) = ⟨ 1, − 2, 5⟩ .
(2.4) HMW:Additional Exercises.
Assume that a wagon is pulled horizontally by an exercising force of 5 lb on the handle at an
angle of 45 with the horizontal.
(a) Illustrate the problem using a rough sketch.
(b) Determine the amount of work done in moving the wagon 30 lb.
48
, MAT1503/101/0/2023
(2.5) HMW:Additional Exercises.
Let the vector ⃗v = ⟨ 3500, 4250⟩ gives the number of units of two models of solar lamps
fabricated by electronics company. Assume that the vector ⃗a = ⟨ 1008.00, 699, 99⟩ gives the
prices (in Rand-ZA) of the two models of solar lamps, respectively.
(a) Calculate the dot product of the two vectors ⃗a and ⃗v .
(b) Explain the meaning of the resulting answer you obtain in the question above.
(c) Let assume that the price of original price of the solar lamps has decreased by 10%.
Identify the vector operation used for this case.
(2.6) HMW:
The force exerted on a rope pulling a toy wagon is30 N. The rope is 30 above the horizontal.
(a) Illustrate the problem by means of a sketch.
(b) Determine the force that pulls the wagon over the ground.
(2.7) Show that there are infinitely many vectors in R3 with Euclidean norm 1 whose Euclidean (2)
inner product with < − 1, 3, − 5 > is zero.
(2.8) Determine all values of k so that ⃗u = < − 3, 2k ,− k > is orthogonal to ⃗v = < 2, 52 , − k > . (1)
Question 3: 5 Marks
(3.1)
(a) Find a and b such that − 3ai − (− 1 − i)b = 3a − 2bi. (1)
(b) Let z1 = 12 + 5i and z2 = (3 − 2i)(2 + λ i ). Find λ without resorting to division such that (1)
z2 = z1 .
(3.2) Let z = − 2 + 3i and z ′ = 5 − 4i. Determine the complex numbers
(a) z 2 − zz ′ . (1)
(b) 1
2 (z
− z) 2 . (1)
(c) 1
2 [z
− z] + [(1 + z′ )]2 . (1)
Question 4: 24 Marks
(4.1) Determine the complex numbers i − 2668 and i − 345 . (2)
h √ i
−i
(4.2) Let z1 = − 1+i , z2 = 11+i 1
− i and z3 = 10 2(i
− 1)i + (− i + 3 − −
3) + (1 i )(1 i ) . (3)
Express z1zz3 2 , z1zz3 2 , and zz3 z1 2 in polar forms.
49
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