What Is the Correct Angle Between Two Vectors If Their Cross Product Is Known?

[Tarrok]

Homework Statement

Two vectors A and B have magnitude A = 3.00 and B = 3.00. Their vector product is A x B= -5.00k + 2.00i. What is the angle between A and B?

Homework Equations

Magnitude of vector product = magnitude of A * magnitude of B * sin of the smaller angle between A and B
|C|=|A||B|*sinX

The Attempt at a Solution

|C|=(5^2+2^2)^(1/2)=3*3*sinX
sinX=(5^2+2^2)^(1/2)/9
X = arcsin[(5^2+2^2)^(1/2)/9]=36.75 degrees

According to the answer, the angle is 37 degrees. As shown above, I see where this comes from. However, we know that sin(36,75)=sin(143,25).

So the magnitude of vector C, where vector C is the cross product of vectors A and B is
3*3sin(36,75)=3*3sin(143,25)=5,385

My question is:
Why is 37 degrees the only correct answer if 143 degrees also works?

Thanks for help

 

[gneill]

Hmm. Good question. I think you have a point. There isn't enough information given about the original vectors A and B to deduce anything about their relative orientation other than the plane that they lie in. So reversing the direction of either A or B will yield a cross product using the supplementary angle (and reverse the direction for the cross product of course). But there's no way to distinguish say, A from -A, or B from -B, or the orientation that the cross product "should" have.

So I'd say that both answers should be acceptable.

I welcome corrections to my thinking here...