The first test will be based on the questions answered here. They will likely come in the form of objective questions covering the same scope.

You are expected to practice these and let me know if there are things you still do not understand by posting specific questions below.

[gview file=”http://oafak.com/wp-content/uploads/2016/02/QA-Vectors-and-Linear-Spaces.pdf”]

Please endeavour to discuss with others and make an effort before posting a question. After a good effort, all questions are welcomed.

Thank You Sir, we appreciate.

Thanks Prof. Great effort.

In the Q&A Number3. second to the last line was not clear. how come about(1-a).

Just look at the line prior to the one you are complaining about: remove “= bxv” and see how the rest of the equality when all terms are placed on one side leads to the equation you seem not to understand!

Thanks Prof.I really appreciate your effort.

Sir, in Q & A number 4(c), we do not have ‘n’ as one of the index but we have it in the answer.Shed more light Sir.

By the reciprocity rule, ⋅=, we have that, 1⋅1=1, 1⋅2=0, 1⋅3=0. It follows that 1 must be perpendicular to the plane of 2and 3 making it parallel to 2×3 A scalar constant must exist such that, 1= 2× 3. I understand the reciprocity rule but the statement that g` is perpenducal g2 and g3 and that is also parallel to the two bases is not clear. Understanding this will help me in getting Q27 and Q29 better. Let someone shed light on this. Thanks.

What you really need to ponder over is the fact that the cross product of two vectors is another vector that MUST be perpendicular to the two vectors! Think about it and your confusion will clear immediately you understand that! That definition is in the most elementary vector book!

Thank you prof.

Let somebody put me through. On page 76 of the new lecture note, I don’t understand how partial differentiation of the position vector ( r) with respect to the new coordinates (r,o z )generate the result of cosoi+ sinoj)dr+ -rsino+ rcoso)do + kdz so as to define the natural bases. thanks.

Thank you everyone. I think I now understand it now.

Happy to see that you sorted yourself out.

good evening sir,you said that the dot product of two scalars will give another scalar,but the dot product of two vectors will give a scalar.please sir explain. Also in vector operation or equations when can we introduce a scalar

Your comment starts rather lazily! “You said, …” is not a good way to start a question because there is no knowledge here that I invented. In fact, I did NOT say what you credited to me! And, remember, this is NOT a first course in vector analysis but we started with vectors as a helpful review!

There is no dot product between two scalars! There is a dot product between two vectors that results in a scalar value!

Between two vectors, three kinds of products exist: 1. dot product (also called scalar or inner product) 2. Vector product (also called cross product) and 3. Tensor product (also called outer, Kronecker or direct product or a dyad)

You are expected to be familiar with the first two products before taking any degree in engineering. It is not my intention to teach this concept but simply to remind you. You can consult ANY elementary book on the subject and you can be sure that AT LEAST those two products will be discussed there. It is your duty to understand these as I will not slow down to teach anyone this particular thing in a postgraduate class. I am only giving you a review here so to let you know that we will build on that knowledge.

The third product – a tensor product (or outer or direct product) that creates a tensor called a dyad is an advanced issue. I will introduce that at the appropriate time.

Your understanding of the earlier products is necessary for you to be able to understand the more advanced concept!

Good day Sir, please Sir, does the tensor product of a vector and itself become 1. In question 34, for instance,the expression,

[e,eXu,eXv]=(uXv).(eOe)e

Reduces to,

=(uXv).e

Please Sir, note that I used O to signify the tensor product here because my device doesnt give me the real expression.

Please ignore that question for the purpose of the test on Tuesday. If you really want to know the definition of the tensor product, refer to the notes on Tensor Algebra. You can find it under SSG805 just like the present notes.

Okay Sir, thank you Sir.

Prof., good day & happy weekend Sir. I have made several efforts to decipher right solutions to some questions and expressions from the given Question & Answer for First Test module, as well as from the slide, but some of them have become quite enigmatic. Although the solutions are there and appeared simple but in actual ‘working’ they are not. Pls Sir I need further explaining on the following:

(i) Question 5: How did you arrive at

11 = ∭ (2 + 2) (, , ) from the given = ∭ ( − ) (1, 2, 3)123; where = 1, = 2, = 3. And also others: 12, 13,22,33. But if I can understand the technique of getting 11, I’ll get others.

(ii) Question 6: How did the magnitude of i.e become 1 (unity)?

(iii)Question 17: The method of solution employed here is quite different than the one used in the note (slide)on page 88. I prefer this method to the one used in the Question & Answer, except that I don’t understand why the 1st & 2nd minor each multiplied by & respectively instead of & , considering the rules of matrix determinant. Plz Sir elucidate this.

(iv) ∙ × = = √. From this expression, what’s than ? I know is a vector not necessarily unit vector as used by examples in the slide & Q&A; but what’s ?

(V) Finally, Sir, is it correct to take it that: subscript equals the inverse of superscript ?

Thank You Sir.

Q5. All you need remember after you have substituted specific values for i and j is that the repetition of an index, such as m, signifies a summation. The rest is straightforward.

Q6. If it is not obvious to you that the values in this question equal unity, then your problem is with elementary trigonometry – SS3 level. You will need to quickly go revise that to cope in this course. A quick review can be found at http://www.cut-the-knot.org/WhatIs/WhatIsTrigonometry.shtml

It is NOT correct to say the superscript equals inverse of subscripts. Such relations occur in the special case of orthogonal systems but is not general. What is always true is the reciprocity relation that occur between the base vectors.

Thank you Sir.

Sir, I have run through the site, can’t see where I can download material for tensor algebra or has it not be uploaded yet? Best regards.

The slides for this section have been posted.