Questions of calculus of crystal structures

[mysci]

We know the rule of cross product

or

Why here |absinΘ| =

, and

= c cos Φ in the above picture?

Thanks for explanation.

 

[Mark44]

View attachment 79881

We know the rule of cross productView attachment 79882 or

Why here |absinΘ| = View attachment 79883 , and View attachment 79884 = c cos Φ in the above picture?

|a||b|sinθ is the area of the base, and is also the magnitude of the cross product of ##\vec{a}## and ##\vec{b}## -- i.e., |##\vec{a}## X ##\vec{b}|##. |c|cosφ gives the height of the parallelipiped. The product of the area of the base and the height gives the volume of the cell.

 

[HallsofIvy]

The base is formed by two vectors with lengths ||a|| and ||b|| and angle between them [itex]\theta[/itex]. Draw a line from the tip of line b to the base, the line forming the vector a. That gives you a right triangle with hypotenuse of length ||b||. So the "opposite side", the height of the parallelogram forming the base of the figure. The "opposite side over the hypotenuse" is sine of the angle so [itex]height/||b||= sin(\theta)[/itex] and [itex]height= ||b|| sin(\theta)[/itex]. The area of a parallelogram is "height times base" so [itex]||a||||b|| sin(\theta)[/itex].

 

[mysci]

|a||b|sinθ is the area of the base, and is also the magnitude of the cross product of ##\vec{a}## and ##\vec{b}## -- i.e., |##\vec{a}## X ##\vec{b}|##. |c|cosφ gives the height of the parallelipiped. The product of the area of the base and the height gives the volume of the cell.

Thanks.

Yes, but why not |axb|, is |axb|(unit vector n) in third step?

absinΘ and |axbl are also magnitudes, but |axb|(unit vector n) is a vector. absinΘ = |axb| ≠ |axb|(unit vector n) = vector a x vector b.

However, here absinΘ = |axb|(unit vector n). I don't understand this.

On the other hand, ccosφ is the height of parallelogram, how to change it to vector c? I don't understand it as well.

Thanks.

 

[Mark44]

Thanks.

Yes, but why not |axb|, is |axb|(unit vector n) in third step?

|a x b| is a scalar, while |a x b|n is a vector that points straight up, and that whose magnitude is the area of the base. If you dot this vector (|a x b|n) with c, you get the volume. One definition for the dot product of a and b is ##a \cdot b = |a| |b| cos(\theta)##, where ##\theta## is the angle between the two vectors. In your problem, the angle is ##\phi##.

absinΘ and |axbl are also magnitudes, but |axb|(unit vector n) is a vector. absinΘ = |axb| ≠ |axb|(unit vector n) = vector a x vector b.

However, here absinΘ = |axb|(unit vector n). I don't understand this.

On the other hand, ccosφ is the height of parallelogram, how to change it to vector c? I don't understand it as well.

Thanks.

 

[mysci]

I may get something.
In fact,
absinΘ = |axb|
ccosΦ = n·c
Is it right?

I thought absinΘ = |axb|n and ccosΦ = c before I get the above thinking.

 

[mysci]

|a x b| is a scalar, while |a x b|n is a vector that points straight up, and that whose magnitude is the area of the base. If you dot this vector (|a x b|n) with c, you get the volume. One definition for the dot product of a and b is ##a \cdot b = |a| |b| cos(\theta)##, where ##\theta## is the angle between the two vectors. In your problem, the angle is ##\phi##.

By the way, how do you type the vector symbol in here? I can't find this symbol. Thanks.

 

[Mark44]

I use LaTeX. Put either two # symbols at the front and two more at the end (for inline) or two $ symbols front and back (for standalone).

Here I'm adding an extra space between each pair so you can see what it looks like without being rendered: # #\vec{a}# #
Removing the spaces gives ##\vec{a}##

 

[mysci]

I use LaTeX. Put either two # symbols at the front and two more at the end (for inline) or two $ symbols front and back (for standalone).

Here I'm adding an extra space between each pair so you can see what it looks like without being rendered: # #\vec{a}# #
Removing the spaces gives ##\vec{a}##

Thanks.

Then
I got following,
absinΘ = |axb|
ccosΦ = n·c
Is it right?

 

[Mark44]

Thanks.

Then
I got following,
absinΘ = |axb|

Should be |a||b|sinθ = |a x b|. a and b are vectors, so ab is not defined. Both sides of the equation should be scalars, which is why you have the magnitudes (absolute values).

ccosΦ = n·c

The right side is a scalar because it's a dot product, so the left side needs to be a scalar as well.
The left side should be |c|cosΦ.

Is it right?

 

[mysci]

Should be |a||b|sinθ = |a x b|. a and b are vectors, so ab is not defined. Both sides of the equation should be scalars, which is why you have the magnitudes (absolute values).
The right side is a scalar because it's a dot product, so the left side needs to be a scalar as well.
The left side should be |c|cosΦ.

Thank you.