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Lecture 2

1. Look at homework problem 2. Look at coordinate systems—Chapter 2 in Massa. Lecture 2. Homework Problem 1. 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

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Lecture 2

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

  2. Homework Problem 1

  3. Optical Crystallography

  4. Polarizing Microscope

  5. 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 · v2 = |v1| x |v2| x cos θ where theta is the angle between the vectors

  6. 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

  7. Right Handed Coordinates

  8. Cartesian Coordinates • |a|, |b|, and |c| all equal 1 • α, β, and γall 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!

  9. v1=(x1,y1,z1) v1· v2 = |v1| x |v2| x cos(θ)=x1x2 +y1y2+z1z2 v · v= |v| x |v| x cos(0)=x2 +y2+z2 |v| = (x2 +y2+z2)1/2 Some Useful Facts

  10. Non-Cartesian Coordinates

  11. 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

  12. 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)‏

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

  14. Assume |a|, |b|, and |c| not equal to one. Assume the vectors are not orthogonal a·b=x1x2|a|2+y1x2|a||b|cosγ+z1x2|a||c|cosβ+ x1y2|a||b|cosγ+y1y2|b|2+z1y2|b||c|cosα+ x1z2|a||c|cosβ+y1z2|b||c|cosα+z1z2|c|2 Dot Product in random coordinates

  15. 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 Magnitude of a Vector

  16. Homework Problem #2

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