Finding Angles with Dot Product: A Beginner's Guide

[Amil]

I'm very new to this. I know that to find the angle between two vectors we use the dot product, but what about finding the angle between a vector and each of it's axes?
for example:
A = 3i+2j+k,

How can I employ the dot product when there is no other vector?

Thanks and keep posting!

Amil

 

[poolwin2001]

Think about this
what are these 3 vectors:
(i)i+0j+0k
(ii)0i+j+0k
(ii)0i+0j+k

 

[Amil]

Please Help!

Those are three vectors who each have two values of zero and a value of 1 along the x (1,0,0), y (0,1,0), and z (0,0,1) axes respectively. Other than thak I don't know. I've been up all night trying to figure this out. I'm really hoping someone can provide me with an example for solving this. I've always worked best through following exaples, however there are no examples of this type of question in my textbook and I've just entered my 28th hour without sleep. PLEASE HELP!

 

[arildno]

Those a three vectors who each have two values of zero and a value of 1 along the x (1,0,0), y (0,1,0), and z (0,0,1) axes respectively.

So what's your problem with using them?
Please make your doubts explicit.

 

[ZapperZ]

Those a three vectors who each have two values of zero and a value of 1 along the x (1,0,0), y (0,1,0), and z (0,0,1) axes respectively. Other than thak I don't know. I've been up all night trying to figure this out. I'm really hoping someone can provide me with an example for solving this. I've always worked best through following exaples, however there are no examples of this type of question in my textbook and I've just entered my 28th hour without sleep. PLEASE HELP!

What poolwin has given you are 3 vectors along the x, y, and z-coordinate axes respectively.

Maybe I suggest that you SKETCH out the problem, and sketch out the vectors that poolwin has given. If you still can't solve this, then there is a more fundamental problem that is involved here and may require that you make a few steps backwards and figure out why.

Zz.

 

[marlon]

You have given this vector 3i + 2j + k

In order to calculate the angle with the x-axis, you need to calculate the dot-product between the given vector and the vector that denotes the x-axis = i + 0j + 0k or
(1,0,0)

So you have (3,2,1)*(1,0,0) = 3*1 + 2*0 + 1*0 = 3
The dot-produvt between two vectors is also defined as
length of vector A * length of vector B * cosinus of the angle between them.

in order to calculate the length of a vector you calculate the squareroot of the dot-product of a vector with itself : length of (3,2,1)=sqrt(3²+2²+1²)=sqrt(14).

So the dot-product yielding 3 is also equal to sqrt(14)*sqrt(1)*cos(x) and x is the angle between the given vector (3,2,1) and the x-axis (1,0,0).

So we have that 3 = sqrt(14)*cos(x) or cos(x)=3/sqrt(14). calculating arcuscosinus yields x =36.6992252°


The same procedure for y and z angles


regards
marlon

 

[Gokul43201]

Marlon, I wish you'd left Amil some work to do, other than translating from Latin (arcuscosinus).

 

[marlon]

Marlon, I wish you'd left Amil some work to do.

Ok, you are right,...maybe I gave away a little bit too much.

I do think that the above answers wouldn't really have helped him out...sorry...

regards
marlon, who gives away far too much

 

[Kelju Ivan]

Maybe it's not too late to do some creative censorship. .)

 

[ZapperZ]

I do think that Gokul brought up a very important aspect of "helping" people on here. None of us should be doing the actual solving of the problem. This does more harm than good, mainly because (1) it gives the impression to the instructor of this student that he/she managed to solve this and so, there's no problem with understanding of the material (2) it washes out the ability for us (and the student) to pin-point where exactly is the source of the problem that is hindering the ability of the student to complete it.

I think that last part is the most crucial factor in helping anyone - figuring out where exactly is the problem. Most of the time, it's the math. Second most common: the inability to know where to start. It is vital that the souce of the problem be narrowed down and addressed immediately, at least by the student him/herself. This, to me, is the most valuable aspect of helping someone in doing any homework problem.

Zz.

 

[marlon]

I think the best way to help someone out is not to give some clues and let the student go on by himself. The best type of help would be to provide him/her with an example of how the basic-systems work and can be applied. just quoting what those basic systems are is most of the time useless information that the student may already know. An instructor needs to provide all kinds of examples in different situations and the work of the student will be :

1) trying to understand the given examples
2) look at them critically and asking the question why and with what goal is a certain formula used

3) this is the most important one : recongize certain patterns and algorithms as the solution goes on... I mean both in math as in physics there are a limited amount of possible ways in which you can solve all kinds of problems. The student needs the see and "feel" what algorithms are usefull for some problem and what are not. A great way to check this out is to say the following : give a student 7 integrals and say he/she must only solve 5 of them. In this case the students need to find out (on first impression) which integral will be the easiest to solve. This is what I mean with knowing the different possible algorithms...

Another great way to teach this ability is to provide multiple answers for one same problem and let the students find out which one is the most efficient solution. or give a question and the "wrong" solution and ask the students what the error in the solution is and why. Then the must print this info in their memory. That is teaching, saying along which axis the x-vector is aligned is NOT...


Just my opinion gained out of tutoring students at college...
regards
marlon

 

[Gokul43201]

Marlon, I agree that having students look at and understand solved examples is a very useful and effective teaching tool.

Nevertheless, I'm sure the students that come here for help have texts that provide them with examples. The least we expect of the students is that they put in the effort to go through what's in the text/class notes before coming here.

From my experience here, I know this is true only with the minority. Far too many students use this forum as the easy way out. And far too many members (I don't mean you) will post complete solutions when the student (I don't mean Amil) has really done very little himself/herself.

Because of the really high traffic in this subforum, I believe PF needs a lot more Mentors here, who can police the posts for abuse.

 

[marlon]

Gokul,

I agree with you on the abuse-thing...
I do think that in this case (i mean this question) the problem is so basic that i think a complete explanation is necessary since the student does not know the basic properties of vector-calculus. Now, we can keep on posting tips on how to solve these problems but if the basics are not well understood this is all in vain. this is the reason I gave such a complete explanation

Just my opinion

regards
marlon

 

[poolwin2001]

I do not agree with pedadogical method suggested by marlon
In my opinion

3) this is the most important one : recongize certain patterns and algorithms as the solution goes on... I mean both in math as in physics there are a limited amount of possible ways in which you can solve all kinds of problems. The student needs the see and "feel" what algorithms are usefull for some problem and what are not.
marlon

Doesnt this set the thinking pattern of the student??
We may be helping some students ,but we will be endangering most would be geniuses.

I think one derives a pleasure(absolute bliss!) from solving a problem espesially after sweating a lot(I do!) this is not lessened by subtle hints but completely blown away by complete solutions.

Just airing out my humble opinion(view,belief,judfement,attitude)
regards
Poolwin2001