# Why should the scalar product of the components along two axes remain unchanged after transformation?

Chemistry Asked by Nimanyu Joshi on October 5, 2021

A couple of proofs from Chapter 2 of "Space Groups for Solid State Scientists" are giving me a hard time (see attached image).

So what I understand is:
"r" was the original lattice point. "r’" is the lattice point r after the mirror operation, which should also be a point of the original lattice. I also have no trouble with how the transformation r->r’ works, and how dot products being zero imply a right angle between the vectors.

So why should the two dot products be equal? And how should one interpret this dot product intuitively (component of one axis onto another?) ? (I can kind of visualize how the mirror image of a triclinic lattice wouldn’t coincide with the original if one of the axes is not perpendicular to the other two, but how should one prove it rigorously, and how does the aforementioned proof work)

The scalar product depends on the length of the vectors and the angle between them. Rotations, translations and mirror operations do not change distances or angles (i.e. don't distort objects), so the scalar product is unchanged as well. There are other operations that do (stretching, warping, etc).

The proof

Without knowing the context, the proof seems wrong. Equation [2.18] is not general and only works for orthogonal coordinate systems. To give a simple example: if the b axis is not perpendicular to the c axis, and we rotate on a twofold along c, the b axis will not be transformed into -b, but rather into a linear combination of b and c. What this proof shows is that if axes are orthogonal, they are orthogonal (tautology).

What is true is that if there is a two-fold axis along c, you can always choose the two other axes such that they are orthogonal to c. This is a choice, however, and other choices are possible.

In the figure, b and c are not perpendicular. Points at the origin, at the tip of b and the tip of c are equivalent by translation. If there is a crystallographic twofold rotation along c, there is another equivalent point at the tip of the rotated b. The rotated b is not the same as negative b, as the figure illustrates by counter-example. However, if you want perpendicular axes (and that is the choice recommended by the international tables), you can switch from the axis b to the thin line (shown as dot-dash-dot-dash), giving you a new axis perpendicular to c. In the case shown, the lattice is now face centered.

Answered by Karsten Theis on October 5, 2021

## Related Questions

### Stubborn CuAAC reaction as derivatization for ESI-MS

1  Asked on January 18, 2021

### Common ion effect

0  Asked on January 16, 2021 by satwik

### Rate of solvolysis of allyl and alkyl halides

1  Asked on January 14, 2021

### Is there a difference between oxidation state and valency?

1  Asked on January 14, 2021 by isaac_samuel

### Bond angles in NH3 and NCl3

3  Asked on January 11, 2021 by kirti-agrawal

### How do I get the smell of Febreeze (uncomplexed cyclodextrin composition) out of my carpet?

0  Asked on January 9, 2021 by annette-r

### Why does magnesium bromide transfers from a carbon atom to nitrogen in piperidine?

1  Asked on January 9, 2021

### Calculation of rupture force in a covalent bond

1  Asked on January 8, 2021 by donghwi-min

### Alphanumeric preference between hexyl and heptyl

1  Asked on January 6, 2021

### Hydrogen venting: Why no flame arrestors?

2  Asked on January 5, 2021 by curious_cat

### Negative DBE/Degree of Hydrogen Deficiency?

0  Asked on January 4, 2021 by uchuuko

### How can we tell if a substitution reaction will be thermodynamically favoured

0  Asked on January 2, 2021 by ali-rizwan

### Dynamic viscosity of gas mixture (appropriate estimation)

0  Asked on December 31, 2020 by josh-e

### My Misconception of Entropy

1  Asked on December 31, 2020 by dval98

### Calcium chloride + starch desiccant packs inside PPE for ebola workers?

0  Asked on December 30, 2020 by jason-holt

### Why is maximum buffer capacity for some diprotic buffers not when pH = pKa?

1  Asked on December 29, 2020 by yeryeon-seo

### Using Beer-Lambert law to calculate light intensities

1  Asked on December 23, 2020 by lblindner

### Converting bromobenzene to 1‐bromo‐3‐methylbenzene

1  Asked on December 23, 2020 by rover

### Coordination compound nomenclature for [Cu(acac)(bpy)]BPh4

0  Asked on December 21, 2020 by marina-calder

### Why are alkali metal compounds like sodium hydride or sodium amide strong bases but weak nucleophiles?

1  Asked on December 19, 2020 by normoorgo

### Ask a Question

Get help from others!