(a) How do the components of a vector5 transform under a translation of coordinates (= x, = y- a, ...

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(a) How do the components of a vector5 transform under a translation of coordinates (= x, = y- a, = z, Fig. 1.16a)? (b) How do the components of a vector transform under an inversion of coordinates...

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(a) How do the components of a vector⁵ transform under a translation of coordinates (= x, = y- a, = z, Fig. 1.16a)?

(b) How do the components of a vector transform under an inversion of coordinates (X= -x, y = -y, z = -z, Fig. 1.16b)?

(c) How do the components of a cross product (Eq XXXXXXXXXXtransform under inversion? [The cross-product of two vectors is properly called a pseudo vector because of this "anomalous" behavior.] Is the cross product of two pseudo vectors a vector, or a pseudo vector? Name two pseudo vector quantities in classical mechanics.

(d) How does the scalar triple product of three vectors transform under inversions? (Such an object is called a pseudo scalar.)

Answered Same Day Dec 25, 2021

Robert · Accepted Answer

x̄=x, ȳ=y−a, z̄=z
It is easiest to look at a real simple example first. A⃗=0 x̂+5ŷ+0ẑ  corresponds to
only a point (5) along the y axis. The particular transformation for y given above
moves y along the +y axis by +a units. Suppose that a was exactly 5. then in the
transformed system, we would have:
⃗̄A=0 ̂̄x+0 ̂̄y+0 ̂̄
So the rule for transforming by translation is this:
⃗̄A=(Ax−ax) x̂+(Ay−ay) ŷ+(Az−az) ẑ
for a generalized translation along vector given by:
a⃗=ax x̂+ay ŷ+azẑ
In the present problem, for a single translation along y, the result is:
⃗̄A=xx̂+(y−a) ŷ+z ẑ
The results (answer) is in fact that the vector is not invariant under translations.
However,  consider  that  A  represents  velocity.  Then  no  matter  where  the
coordinates are translated, each observer would see the same velocity. However,
if A represents position, then in fact the components of A would change. Position
seems to be a unique type of vector in this regard; however position does not
often appear in physics, instead we talk about change in position or also directed
distance.
A simple yes or no here does not suffice actually.
“Be very careful about distinguishing between points and vectors. Otherwise you
are likely to get bitten by translations doing the Wrong Thing.”
⃗̄A=−a⃗+A⃗ : d
⃗̄A
dt =−
da⃗
dt +
dA⃗
dt =0+
dA⃗
dt =
dA⃗
dt
Quantities  derived  from  position  vectors  are  invariant  under  a
translation. However a position vector itself is not invariant under a translation
which some interpret as meaning that position is, in fact, a slightly different type
of vector than other vectors. However I would contend that the velocity vector
would also not be invariant under a translation at a constant velocity.
To really understand this completely, we need to look at the definition of a vector:
it has magnitude and direction (and is not invariant under inversion). So while the
coordinates of a point change under translation, the distance between two points
does  not  change  under  translation.  So  when  we  speak  about  a  vector  in  n-
dimensional space, and although the vector is normally written as if it were only
those n-coordinate points,

(a) How do the components of a vector5 transform under a translation of coordinates (= x, = y- a, = z, Fig. 1.16a)? (b) How do the components of a vector transform under an inversion of coordinates...

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