Dot Product/Cross Product Interpretation, Geometric Constructionby dr721 Tags: construction, geometric, interpretation, product, product or cross 

#1
Sep412, 11:16 PM

P: 23

1. The problem statement, all variables and given/known data
Given the nonzero vector a ε ℝ^{3}, a[itex]\dot{}[/itex]x = b ε ℝ, and a × x = c ε ℝ^{3}, can you determine the vector x ε ℝ^{3}? If so, give a geometric construction for x. 2. Relevant equations a[itex]\dot{}[/itex]x = axcos[itex]\Theta[/itex] 3. The attempt at a solution I'm not really certain what it is asking for? Obviously, the cross product of the two vectors creates a vector perpendicular to all vectors in the a, x plane. And, the magnitude of the cross product defines the area of a parallelogram spanned by a and x. Also, xcos[itex]\Theta[/itex] is the length of the projection of x onto a, which is also equal to b/a But while I know all this, I don't know what I'm trying to show or how to show it? Any help would be great! Thanks! 



#2
Sep412, 11:36 PM

P: 834

Let's say you know the vector [itex]a[/itex] and all of its components. You know its dot product and cross product with [itex]x[/itex]. Can you use this information to actually figure out what each of the components of [itex]x[/itex] should be? Can you describe a picture in which it's clear how to figure out what [itex]x[/itex] should be?




#3
Sep412, 11:48 PM

P: 23

I know a lot of tedious algebra might describe it, but I guess I don't really know how I would describe a picture per se. I don't know how I would break it down to each of its components. 



#4
Sep412, 11:54 PM

P: 834

Dot Product/Cross Product Interpretation, Geometric Construction
Right, it's not something that's immediately obvious in an an abstract sense, so let me rephrase the question.
If you have a point on the unit circle and you know both the sine of the angle it makes with the +x axis and the cosine as well, do you know the point's location on the unit circle absolutely? Do you have a formula for the cross product that involves either sine or cosine? 



#5
Sep512, 12:03 AM

P: 23

Yes, it would stand to reason that you would know that point.
And no, we have yet to learn such a formula. 



#6
Sep512, 12:12 AM

P: 834

Hm, without knowing that [itex]a \times x = a x \sin \theta[/itex] where [itex]\theta[/itex] is the angle between the vectors, I think it would be difficult to make the required connection.
Let's imagine for a moment that [itex]a,x[/itex] lie in some plane. It stands to reason that [itex]c[/itex] as defined in the problem is perpendicular to this plane. Obviously you don't know [itex]x[/itex], but you should be able to find some other vector in the plane just with [itex]a[/itex] and [itex]c[/itex]. In particular, if you use a cross product to do this, you'll know that the result will be perpendicular to [itex]a[/itex] yet still in the plane [itex]a,x[/itex] span. Once you're at this point, where you have two perpendicular vectors and [itex]x[/itex] must lie in the plane that they define, you should see that this is just a unit circle problem in some arbitrary plane. The dot product gives you the cosine, and the cross product gives you the sine. 



#7
Sep512, 12:27 AM

P: 23

Well, that formula appears nowhere in my notes, memory, or book up to this point. So I would assume he wants us to be able to solve the problem without it. Would that be possible?
Otherwise, crossing a with c would give me a vector y that lies in the a, x span. But then how are you applying the two formulas to determine the components of x? Are you using a and y? 



#8
Sep512, 06:02 AM

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That formula is one you should recognize as the area of the parallelogram formed by a and x.
It might help you to visualize the problem if you orient your coordinate system so that a points along the xaxis and x lies in the xyplane. 



#9
Sep512, 07:20 AM

P: 23

Right, I can see the formula as being true. My fear was that as he had yet to introduce it in class, my professor may frown upon its use.
That said, I'm a bit confused about this idea of creating another vector in that same plane and using it to solve for x. I don't see how the formulas allow for that. 



#10
Sep512, 12:34 PM

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I wouldn't worry about using the formula. In fact, he might consider it something you should be able to deduce using basic trig and geometry from the fact that the magnitude of c is the area of the parallelogram.



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