Lecture 2

1. Look at homework problem2. Look at coordinate systems—Chapter 2 in Massa

Homework Problem 11. It was well known that there were two crystalline forms of sulfur long beforethe descovery of x-rays in 1895. How were the space group classes determined without x-rays?

Answer: Optical Goniometerand Optical Crystallography

Optical Crystallography

Polarizing Microscope

Vectors

Vectors have length and direction We will use bold v to represent a vector |V| is the magnitude (length) of a vector The dot product of two vectors is as follows

v1 · v

2 = |v

1| x |v

2| x cos θ where theta is the

angle between the vectors

Coordinates and Basis Vectors

A coordinate system is composed of three basis vectors call a, b, and c.

The angles between these vectors are given by α (alpha) the angle between b and c β (beta) the angle between a and c γ (gamma) the angle between a and b

Right Handed Coordinates

Since space has no preferred arrangementthere are two coordinate systems possible.Always use a right handed system!

Cartesian Coordinates

|a|, |b|, and |c| all equal 1 α, β, and γ all 90º A reminder cos(0)=1 cos (90)=0 sin(0)=0

sin(90)=1 The symbols x, y, and z represent positions

along a, b, and c A general vector is given by

v=xa + yb + zc Since the basis vectors magnitude is 1 they are

ignored!

Some Useful Facts

v1 = (x

1,y

1,z

1)

v1 · v

2 = |v

1| x |v

2| x cos(θ)=x

1x

2 +y

1y

2+z

1z

2

v · v = |v| x |v| x cos(0)=x2 +y2+z2

|v| = (x2 +y2+z2)1/2

Non-Cartesian Coordinates

• Its cargo compartment is 121 ft long, 13.5 ft high,and 19 ft wide (37 m by 4.1 m by 5.8 m), or justover 31,000 ft³ (880 m³). The compartment canaccommodate up to 36 463L master pallets or amix of palletized cargo and vehicles.

• It has an upper deck seating area for 73passengers.

Let's make the basis vectors the length, width, and height of the cargo compartment.

Therefore |a|=121 |b|=19 and |c|=13.5 The basis vectors are orthogonal Place the origin at the front left of the plane. Now the coordinates inside the plane are

fractional. v=xa + yb + zc

Non-orthogonal Coordinates

If the basis vectors have a magnitude of 1 then a·a = b·b = c·c = 1

a·b = cos(γ) b·c = cos(α) a·c = cos(β)

Crystallographic Coordinate

These can provide the worst of all possible worlds.They frequently are non-orthogonalThe do not have unit vectors as the basis vectors.The coordinates system is defined by the edges of the unit cell.

Dot Product in random coordinates

Assume |a|, |b|, and |c| not equal to one.Assume the vectors are not orthogonal

a·b=x1x

2|a|2+y

1x

2|a||b|cosγ+z

1x

2|a||c|cosβ+

x1y

2|a||b|cosγ+y

1y

2|b|2+z

1y

2|b||c|cosα+

x1z

2|a||c|cosβ+y

1z

2|b||c|cosα+z

1z

2|c|2

Magnitude of a Vector

This can be derived from the dot product formula by making x1 and x2 into x, etc.

|v|2=x2|a|2+y2|b|2+z2|c|2+2xy|a||b|cosγ+

2xz|a||c|cosβ+2yz|b||c|cosα

Homework Problem #2

2. Using the C5 reference coordinates (l=121, w=19, h=13.5) calculatethe distance between 2 packages located at (0.33,0.16,0.45) and(0.52,0.43,0.23).

Are the coordinates (0.85,0.5,1.1) valid?