Vector Transformations: Problem 1.10(a-c) - DJGriffiths

[Living_Dog]

Problem 1.10(a) of DJGriffiths asks: "How do the components of a vector transform under a translation of coordinates?"

This is confusing me (not hard to do) since the translation is given, then isn't it just:

x' = x + A

where A = $$\left(\begin{array}{c} 0 \\ -a \\ 0 \end{array}\right)$$

Problem 1.10(b): The same "... inversion ..." so that x' = -I x?

Problem 1.10(c): "How does the cross-product of two vectors transform under inversion?"

Once again, if A is a vector, then it transforms as always A' = RA.

So how is it any different if the vector is generated by a cross-product or is made up by me? It's a vector! Unless the question is not asking about A', but rather about how does BxC transform? ...how would I apply the ransformation to the actual cross-product? I mean, do I take RBxRC or R(BxC)??

sorry if my questions are annoying or too frequent. but it's always the same thing - I read the chapter and have no problem following the theory. Then I get to the Problems section and suddenly it's like the questions have nothing to do with the chapter I just read!

-LD

 

[quantumdude]

Problem 1.10(a) of DJGriffiths asks: "How do the components of a vector transform under a translation of coordinates?"

This is confusing me (not hard to do) since the translation is given, then isn't it just:

x' = x + A

No, vectors are invariant under translation. For instance if you and I are at rest relative to each other but standing at different locations, and one of us observes a car zipping by at 50 mph due east, then the other of us will agree with that velocity measurement.

Problem 1.10(b): The same "... inversion ..." so that x' = -I x?

Just carry out the matrix multiplication and you'll have your answer.

Problem 1.10(c): "How does the cross-product of two vectors transform under inversion?"

Once again, if A is a vector, then it transforms as always A' = RA.

So how is it any different if the vector is generated by a cross-product or is made up by me? It's a vector!

Strictly speaking the cross product of two vectors is not a vector: It's an axial vector or pseudovector.

Unless the question is not asking about A', but rather about how does BxC transform? ...how would I apply the ransformation to the actual cross-product? I mean, do I take RBxRC or R(BxC)??

Yes, that's what they're asking. You would transform the vectors B and C first, then take their cross product.