Hi divinelogos,
If are two vectors the cross product is defined by,
The direction of is given by the Right hand rule.
Therefore,
Since is perpendicular to ;
It should be noted that, we have to assume that to get the given answer.
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 divinelogos,
If are two vectors the cross product is defined by,
The direction of is given by the Right hand rule.
Therefore,
Since is perpendicular to ;
It should be noted that, we have to assume that 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
This is not true. The length of the vector is
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.)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.
The length of the vector would be . Since the cross product of two vectors is perpendicular to both, in the "right hand rule" direction, and we are given that is perpendiculat to both and , is either or 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, . 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 to , your thumb would be pointing in the direction of but I don't see that explicitely stated.
Any ideas would be appreciated and I will definitely click the little thank you button for you