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I'm trying to help a friend with the following homework question:

Vectors A and B lie in an xy plane. A has a magnitude of 8.00 units and an angle of 130 degrees; B has components Bx = -7.72 units and By = -9.2 units. Find the angle between the negative direction of y-axis and the direction of the product AxB.

Now I have already calculated B have an angle of 230.11 degrees. I can calculate the angle between the two vectors using the dot product method, but the question asked for the angle between the negative y-axis and the cross-product of the two vectors, the cross-product being a vector that is both perpendicular to both A and B, which is causing a problem for me. Can somebody give me some help in resolving this particular issue? It would be much appreciated.

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Think of any vector lying on the $y$-axis (pointing in the negative direction) and calculate the dot product between this and $A\times B$. – draks ... Jul 16 '12 at 22:48
@draks - But wouldn't the cross-product of A and B be normal to the x-y plane? – D Brown Jul 16 '12 at 22:55
right. So the angle is $\pi/2$, as in Ross' answer. – draks ... Jul 17 '12 at 7:04
up vote 3 down vote accepted

The answer to the question as given is indicated in your comment. The cross product of any two vectors in the $xy$ plane (unless they are parallel) is in the $\pm z$ direction. Its angle with the negative $y$ axis (or any vector in the $xy$ plane) is $\frac \pi 2$.

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Thanks Ross!! I guess I was thinking about the question too hard and ignoring some of the principles...thanks for pointing them out. Much appreciated!!! :) – D Brown Jul 16 '12 at 23:26

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