Cross Product Problem

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September 9th 2011, 04:58 PM
divinelogos

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Cross Product Problem

Basically, I've drawn the diagram and considered the fact that the geometric definition of the cross product is: U x V = |u||v|sin(theta).

Given that the angle between them is pi/6, then sin(theta) is equal to 1/2. So, I see where they get the 1/2 in the answer. However, these are my questions:

1. I'm unsure how to relate the 1/2 to the vector (e3) perpendicular to the plane spanned by e1 and e2. How do they relate?

2. Does normal basis mean each of the vectors in the basis is a unit vector and that therefore there magnitude's are all 1? This would simplify the problem but i'm not sure if it's correct.

Any ideas would be appreciated and I will definitely click the little thank you button for you :) (Hi)
September 9th 2011, 06:36 PM
Sudharaka

Re: Cross Product Problem

Quote:

Originally Posted by divinelogos

Basically, I've drawn the diagram and considered the fact that the geometric definition of the cross product is: U x V = |u||v|sin(theta).

Given that the angle between them is pi/6, then sin(theta) is equal to 1/2. So, I see where they get the 1/2 in the answer. However, these are my questions:

1. I'm unsure how to relate the 1/2 to the vector (e3) perpendicular to the plane spanned by e1 and e2. How do they relate?

2. Does normal basis mean each of the vectors in the basis is a unit vector and that therefore there magnitude's are all 1? This would simplify the problem but i'm not sure if it's correct.

Correct.

Any ideas would be appreciated and I will definitely click the little thank you button for you :) (Hi)

Hi divinelogos,

If $\underline{a}\mbox{ and }\underline{b}$ are two vectors the cross product is defined by,

$\underline{a}\times\underline{b}=\underline{\hat{n }}|\underline{a}||\underline{b}|\sin\theta~;~\mbox { where }\theta\mbox{ is the angle between }\underline{a}\mbox{ and }\underline{b}~;~\underline{\hat{n}}\mbox{ is the unit vector perpendicular to }\underline{a}\mbox{ and }\underline{b}$

The direction of $\underline{\hat{n}}$ is given by the Right hand rule.

Therefore, $e_{1}\times e_{2}=\underline{\hat{n}}|e_{1}||e_{2}|\sin \frac{\pi}{6}=\frac{1}{2}\underline{\hat{n}}$

Since $e_{3}$ is perpendicular to $e_{1}\mbox{ and }e_{2}$ ;

$\underline{\hat{n}}=e_{3}$

$\therefore e_{1}\times e_{2}=\frac{1}{2}e_{3}$

It should be noted that, we have to assume that $\underline{\hat{n}}\neq -e_{3}$ to get the given answer.
September 10th 2011, 12:35 PM
divinelogos

Re: Cross Product Problem

Quote:

Originally Posted by Sudharaka

Hi divinelogos,

If $\underline{a}\mbox{ and }\underline{b}$ are two vectors the cross product is defined by,

$\underline{a}\times\underline{b}=\underline{\hat{n }}|\underline{a}||\underline{b}|\sin\theta~;~\mbox { where }\theta\mbox{ is the angle between }\underline{a}\mbox{ and }\underline{b}~;~\underline{\hat{n}}\mbox{ is the unit vector perpendicular to }\underline{a}\mbox{ and }\underline{b}$

The direction of $\underline{\hat{n}}$ is given by the Right hand rule.

Therefore, $e_{1}\times e_{2}=\underline{\hat{n}}|e_{1}||e_{2}|\sin \frac{\pi}{6}=\frac{1}{2}\underline{\hat{n}}$

Since $e_{3}$ is perpendicular to $e_{1}\mbox{ and }e_{2}$ ;

$\underline{\hat{n}}=e_{3}$

$\therefore e_{1}\times e_{2}=\frac{1}{2}e_{3}$

It should be noted that, we have to assume that $\underline{\hat{n}}\neq -e_{3}$ to get the given answer.

Thanks Sudharaka! I have one question about your solution though:

Why can we assume n=e3 if n is a unit vector? Does "normalized basis" in the problem statement mean e1,e2, and e3 are unit vectors as well? If this is the case, your solution looks correct.

Thanks again for your help :)
September 10th 2011, 02:55 PM
HallsofIvy

Re: Cross Product Problem

Quote:

Originally Posted by divinelogos

Basically, I've drawn the diagram and considered the fact that the geometric definition of the cross product is: U x V = |u||v|sin(theta).

This is not true. The length of the vector $U\times V$ is $|U||V| sin(\theta)$

Quote:

Given that the angle between them is pi/6, then sin(theta) is equal to 1/2. So, I see where they get the 1/2 in the answer. However, these are my questions:

1. I'm unsure how to relate the 1/2 to the vector (e3) perpendicular to the plane spanned by e1 and e2. How do they relate?

2. Does normal basis mean each of the vectors in the basis is a unit vector and that therefore there magnitude's are all 1? This would simplify the problem but i'm not sure if it's correct.

Yes, to "normalize" a vector means to divide by its length to get a vector in the same direction with length 1. A "normalized" basis is a basis of vectors, each of length 1. (An "orthonormal" basis would be a basis consisting of vectors,each of length 1, each at right angles to the others- but you are told that the distance between e1 and e2 so this is NOT an orthonormal vector.)

The length of the vector $e_1\times e_2$ would be $|e_1||e_2|sin(\theta)= (1)(1)(1/2)= 1/2$ . Since the cross product of two vectors is perpendicular to both, in the "right hand rule" direction, and we are given that $e_3$ is perpendiculat to both $e_1$ and $e_2$ , $e_1\times e_2$ is either $\frac{1}{2}e_3$ or $-\frac{1}{2}e_3$ but we don't have enough information to decide which. That would depend upon in which of two possible directions, perpendicular to the plane determined by the two vectors, $e_3$ . They are apparently assuming that the three vectors form a "right hand frame"- that is, that if you curl the fingers of your right hand from $e_1$ to $e_2$ , your thumb would be pointing in the direction of $e_3$ but I don't see that explicitely stated.

Quote:

Any ideas would be appreciated and I will definitely click the little thank you button for you :) (Hi)
September 11th 2011, 01:35 PM
divinelogos

Re: Cross Product Problem

Quote:

Originally Posted by HallsofIvy

This is not true. The length of the vector $U\times V$ is $|U||V| sin(\theta)$

I just missed the normal vector on the end of the definition... my bad.
September 12th 2011, 03:48 PM
divinelogos

Re: Cross Product Problem

I understand now how to get e1xe2, but what about e2xe3 and e1xe3?

e1xe2 generates a vector perpendicular to the plane spanned by e1 and e2, but i'm not sure how they got 2e1-rad(3)e2?