Miller indices for the directions inthe hexagonal lattice using three- digit and four-digit systems. Concept introduction: Miller indices at the planes pass through the three points at edges of the crystal structure. In crystalline material, all the points, planes and directions are specified within the unit cell. The direction is defined as a line between two points.

Question

Want to see more full solutions like this?

Answer 1

Question

Chapter 3, Problem 3.57P

Interpretation Introduction

Interpretation:

Miller indices for the directions inthe hexagonal lattice using three- digit and four-digit systems.

Concept introduction:

Miller indices at the planes pass through the three points at edges of the crystal structure. In crystalline material, all the points, planes and directions are specified within the unit cell. The direction is defined as a line between two points.

Expert Solution & Answer

Answer to Problem 3.57P

Miller indices of hexagonal lattice at given directions.

PlaneThree digit systemFour−digit systemA[ 1 ¯ 0 2][ 2 ¯ 1 1 6]B[0 1 ¯ 1 ¯][1 2 ¯ 1 3 ¯]C[ 1 ¯ 1 1 ¯][ 1 ¯ 1 0 1 ¯]

Explanation of Solution

Given:

The following hexagonal lattice:

Essentials Of Materials Science And Engineering, Chapter 3, Problem 3.57P

Steps for calculating miller indices for given directions are:

Find head point coordinate (direction of a head point from the origin (x,y,z) )
Find-tail point coordinate (x,y,z)
Head point coordinate − tail point coordinate

Miller indices for plane A:

Head points = 0,1,1

Tail point = 1/2, 1, 0

Head point − Tail point:

= (−1/2, 0, 1)= (−1/2, 0, 1)×2= (−1, 0, 2)= [1¯ 0 2]

Four-digit system

Convert the three-digitsystem into a four-digit system.

[h,k,l] = [1¯ 0 2]

Coordinates of four-digit system.

h=1/3[2h−k] =1/3[−2−0] =−2/3k=1/3[2k−h] =1/3[0−( −1)] =1/3i=−1/3[h+k] =1/3[−1+0] =+1/3l=−1/3[h+k] =1/3[−1+0] =+1/3

l=l=2[h k i l] = [−2/3, 1/3, +1/3, 2] = [−2/3, 1/3, +1/3, 2]×3 = [−2, 1, +1, 6] = [ 2 ¯ 1 1 6]

Miller indices for plane B:

Head points = 1,0,0

Tail point = 1, 1, 1

Head point − Tail point:

=0, −1, −1= [0 1¯ 1¯]

Four-digit system

Convert the three-digitsystem into a four-digit system.

[h,k,l] = [0 1¯ 1¯]

Coordinates of the four-digit system.

h=1/3[2h−k] =1/3[0−( −1)] =1/3k=1/3[2k−h] =1/3[−2−0] =−2/3i=−1/3[h+k] =1/3[0+( −1)] =1/3l=l=−1[h k i l] = [1/3, −2/3, 1/3, −1 = [1/3, −2/3, 1/3, −1]*3 = [1, −2, 1, −3] = [1 2 ¯ 1 3 ¯]

Miller indices for plane C:

Three digit system:

Head points = 0,0,0

Tail point = 1, -1, 1

Head point − Tail point:

= (−1, 1 −1)= [1¯ 1 1¯]

Four-digit system

Convert three systems into four-digit system.

[h,k,l] = [−1,1,−1]h=1/3[2h−k] =1/3[−2−1] =−1k=1/3[2k−h] =1/3[2+1] =1i=−1/3[h+k] =−1/3[−1+1] =0l=l=−1[h k i l] = [−1, 1, 0, −1] = [ 1 ¯ 1 0 1 ¯]

Conclusion

Miller indices of hexagonal lattice at given directions.

PlaneThree digit systemFour−digit systemA[ 1 ¯ 0 2][ 2 ¯ 1 1 6]B[0 1 ¯ 1 ¯][1 2 ¯ 1 3 ¯]C[ 1 ¯ 1 1 ¯][ 1 ¯ 1 0 1 ¯]

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Answer 2

Question

Chapter 3, Problem 3.57P

Interpretation Introduction

Interpretation:

Miller indices for the directions inthe hexagonal lattice using three- digit and four-digit systems.

Concept introduction:

Miller indices at the planes pass through the three points at edges of the crystal structure. In crystalline material, all the points, planes and directions are specified within the unit cell. The direction is defined as a line between two points.

Expert Solution & Answer

Answer to Problem 3.57P

Miller indices of hexagonal lattice at given directions.

PlaneThree digit systemFour−digit systemA[ 1 ¯ 0 2][ 2 ¯ 1 1 6]B[0 1 ¯ 1 ¯][1 2 ¯ 1 3 ¯]C[ 1 ¯ 1 1 ¯][ 1 ¯ 1 0 1 ¯]

Explanation of Solution

Given:

The following hexagonal lattice:

Essentials Of Materials Science And Engineering, Chapter 3, Problem 3.57P

Steps for calculating miller indices for given directions are:

Find head point coordinate (direction of a head point from the origin (x,y,z) )
Find-tail point coordinate (x,y,z)
Head point coordinate − tail point coordinate

Miller indices for plane A:

Head points = 0,1,1

Tail point = 1/2, 1, 0

Head point − Tail point:

= (−1/2, 0, 1)= (−1/2, 0, 1)×2= (−1, 0, 2)= [1¯ 0 2]

Four-digit system

Convert the three-digitsystem into a four-digit system.

[h,k,l] = [1¯ 0 2]

Coordinates of four-digit system.

h=1/3[2h−k] =1/3[−2−0] =−2/3k=1/3[2k−h] =1/3[0−( −1)] =1/3i=−1/3[h+k] =1/3[−1+0] =+1/3l=−1/3[h+k] =1/3[−1+0] =+1/3

l=l=2[h k i l] = [−2/3, 1/3, +1/3, 2] = [−2/3, 1/3, +1/3, 2]×3 = [−2, 1, +1, 6] = [ 2 ¯ 1 1 6]

Miller indices for plane B:

Head points = 1,0,0

Tail point = 1, 1, 1

Head point − Tail point:

=0, −1, −1= [0 1¯ 1¯]

Four-digit system

Convert the three-digitsystem into a four-digit system.

[h,k,l] = [0 1¯ 1¯]

Coordinates of the four-digit system.

h=1/3[2h−k] =1/3[0−( −1)] =1/3k=1/3[2k−h] =1/3[−2−0] =−2/3i=−1/3[h+k] =1/3[0+( −1)] =1/3l=l=−1[h k i l] = [1/3, −2/3, 1/3, −1 = [1/3, −2/3, 1/3, −1]*3 = [1, −2, 1, −3] = [1 2 ¯ 1 3 ¯]

Miller indices for plane C:

Three digit system:

Head points = 0,0,0

Tail point = 1, -1, 1

Head point − Tail point:

= (−1, 1 −1)= [1¯ 1 1¯]

Four-digit system

Convert three systems into four-digit system.

[h,k,l] = [−1,1,−1]h=1/3[2h−k] =1/3[−2−1] =−1k=1/3[2k−h] =1/3[2+1] =1i=−1/3[h+k] =−1/3[−1+1] =0l=l=−1[h k i l] = [−1, 1, 0, −1] = [ 1 ¯ 1 0 1 ¯]

Conclusion

Miller indices of hexagonal lattice at given directions.

PlaneThree digit systemFour−digit systemA[ 1 ¯ 0 2][ 2 ¯ 1 1 6]B[0 1 ¯ 1 ¯][1 2 ¯ 1 3 ¯]C[ 1 ¯ 1 1 ¯][ 1 ¯ 1 0 1 ¯]

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Answer 3

Question

Chapter 3, Problem 3.57P

Interpretation Introduction

Interpretation:

Miller indices for the directions inthe hexagonal lattice using three- digit and four-digit systems.

Concept introduction:

Miller indices at the planes pass through the three points at edges of the crystal structure. In crystalline material, all the points, planes and directions are specified within the unit cell. The direction is defined as a line between two points.

Expert Solution & Answer

Answer to Problem 3.57P

Miller indices of hexagonal lattice at given directions.

PlaneThree digit systemFour−digit systemA[ 1 ¯ 0 2][ 2 ¯ 1 1 6]B[0 1 ¯ 1 ¯][1 2 ¯ 1 3 ¯]C[ 1 ¯ 1 1 ¯][ 1 ¯ 1 0 1 ¯]

Explanation of Solution

Given:

The following hexagonal lattice:

Essentials Of Materials Science And Engineering, Chapter 3, Problem 3.57P

Steps for calculating miller indices for given directions are:

Find head point coordinate (direction of a head point from the origin (x,y,z) )
Find-tail point coordinate (x,y,z)
Head point coordinate − tail point coordinate

Miller indices for plane A:

Head points = 0,1,1

Tail point = 1/2, 1, 0

Head point − Tail point:

= (−1/2, 0, 1)= (−1/2, 0, 1)×2= (−1, 0, 2)= [1¯ 0 2]

Four-digit system

Convert the three-digitsystem into a four-digit system.

[h,k,l] = [1¯ 0 2]

Coordinates of four-digit system.

h=1/3[2h−k] =1/3[−2−0] =−2/3k=1/3[2k−h] =1/3[0−( −1)] =1/3i=−1/3[h+k] =1/3[−1+0] =+1/3l=−1/3[h+k] =1/3[−1+0] =+1/3

l=l=2[h k i l] = [−2/3, 1/3, +1/3, 2] = [−2/3, 1/3, +1/3, 2]×3 = [−2, 1, +1, 6] = [ 2 ¯ 1 1 6]

Miller indices for plane B:

Head points = 1,0,0

Tail point = 1, 1, 1

Head point − Tail point:

=0, −1, −1= [0 1¯ 1¯]

Four-digit system

Convert the three-digitsystem into a four-digit system.

[h,k,l] = [0 1¯ 1¯]

Coordinates of the four-digit system.

h=1/3[2h−k] =1/3[0−( −1)] =1/3k=1/3[2k−h] =1/3[−2−0] =−2/3i=−1/3[h+k] =1/3[0+( −1)] =1/3l=l=−1[h k i l] = [1/3, −2/3, 1/3, −1 = [1/3, −2/3, 1/3, −1]*3 = [1, −2, 1, −3] = [1 2 ¯ 1 3 ¯]

Miller indices for plane C:

Three digit system:

Head points = 0,0,0

Tail point = 1, -1, 1

Head point − Tail point:

= (−1, 1 −1)= [1¯ 1 1¯]

Four-digit system

Convert three systems into four-digit system.

[h,k,l] = [−1,1,−1]h=1/3[2h−k] =1/3[−2−1] =−1k=1/3[2k−h] =1/3[2+1] =1i=−1/3[h+k] =−1/3[−1+1] =0l=l=−1[h k i l] = [−1, 1, 0, −1] = [ 1 ¯ 1 0 1 ¯]

Conclusion

Miller indices of hexagonal lattice at given directions.

PlaneThree digit systemFour−digit systemA[ 1 ¯ 0 2][ 2 ¯ 1 1 6]B[0 1 ¯ 1 ¯][1 2 ¯ 1 3 ¯]C[ 1 ¯ 1 1 ¯][ 1 ¯ 1 0 1 ¯]

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Answer 4

Question

Chapter 3, Problem 3.57P

Interpretation Introduction

Interpretation:

Miller indices for the directions inthe hexagonal lattice using three- digit and four-digit systems.

Concept introduction:

Miller indices at the planes pass through the three points at edges of the crystal structure. In crystalline material, all the points, planes and directions are specified within the unit cell. The direction is defined as a line between two points.

Expert Solution & Answer

Answer to Problem 3.57P

Miller indices of hexagonal lattice at given directions.

PlaneThree digit systemFour−digit systemA[ 1 ¯ 0 2][ 2 ¯ 1 1 6]B[0 1 ¯ 1 ¯][1 2 ¯ 1 3 ¯]C[ 1 ¯ 1 1 ¯][ 1 ¯ 1 0 1 ¯]

Explanation of Solution

Given:

The following hexagonal lattice:

Essentials Of Materials Science And Engineering, Chapter 3, Problem 3.57P

Steps for calculating miller indices for given directions are:

Find head point coordinate (direction of a head point from the origin (x,y,z) )
Find-tail point coordinate (x,y,z)
Head point coordinate − tail point coordinate

Miller indices for plane A:

Head points = 0,1,1

Tail point = 1/2, 1, 0

Head point − Tail point:

= (−1/2, 0, 1)= (−1/2, 0, 1)×2= (−1, 0, 2)= [1¯ 0 2]

Four-digit system

Convert the three-digitsystem into a four-digit system.

[h,k,l] = [1¯ 0 2]

Coordinates of four-digit system.

h=1/3[2h−k] =1/3[−2−0] =−2/3k=1/3[2k−h] =1/3[0−( −1)] =1/3i=−1/3[h+k] =1/3[−1+0] =+1/3l=−1/3[h+k] =1/3[−1+0] =+1/3

l=l=2[h k i l] = [−2/3, 1/3, +1/3, 2] = [−2/3, 1/3, +1/3, 2]×3 = [−2, 1, +1, 6] = [ 2 ¯ 1 1 6]

Miller indices for plane B:

Head points = 1,0,0

Tail point = 1, 1, 1

Head point − Tail point:

=0, −1, −1= [0 1¯ 1¯]

Four-digit system

Convert the three-digitsystem into a four-digit system.

[h,k,l] = [0 1¯ 1¯]

Coordinates of the four-digit system.

h=1/3[2h−k] =1/3[0−( −1)] =1/3k=1/3[2k−h] =1/3[−2−0] =−2/3i=−1/3[h+k] =1/3[0+( −1)] =1/3l=l=−1[h k i l] = [1/3, −2/3, 1/3, −1 = [1/3, −2/3, 1/3, −1]*3 = [1, −2, 1, −3] = [1 2 ¯ 1 3 ¯]

Miller indices for plane C:

Three digit system:

Head points = 0,0,0

Tail point = 1, -1, 1

Head point − Tail point:

= (−1, 1 −1)= [1¯ 1 1¯]

Four-digit system

Convert three systems into four-digit system.

[h,k,l] = [−1,1,−1]h=1/3[2h−k] =1/3[−2−1] =−1k=1/3[2k−h] =1/3[2+1] =1i=−1/3[h+k] =−1/3[−1+1] =0l=l=−1[h k i l] = [−1, 1, 0, −1] = [ 1 ¯ 1 0 1 ¯]

Conclusion

Miller indices of hexagonal lattice at given directions.

PlaneThree digit systemFour−digit systemA[ 1 ¯ 0 2][ 2 ¯ 1 1 6]B[0 1 ¯ 1 ¯][1 2 ¯ 1 3 ¯]C[ 1 ¯ 1 1 ¯][ 1 ¯ 1 0 1 ¯]

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Answer 5

Chapter 3, Problem 3.57P

Interpretation Introduction

Interpretation:

Miller indices for the directions inthe hexagonal lattice using three- digit and four-digit systems.

Concept introduction:

Miller indices at the planes pass through the three points at edges of the crystal structure. In crystalline material, all the points, planes and directions are specified within the unit cell. The direction is defined as a line between two points.

Expert Solution & Answer

Answer to Problem 3.57P

Miller indices of hexagonal lattice at given directions.

PlaneThree digit systemFour−digit systemA[ 1 ¯ 0 2][ 2 ¯ 1 1 6]B[0 1 ¯ 1 ¯][1 2 ¯ 1 3 ¯]C[ 1 ¯ 1 1 ¯][ 1 ¯ 1 0 1 ¯]

Explanation of Solution

Given:

The following hexagonal lattice:

Essentials Of Materials Science And Engineering, Chapter 3, Problem 3.57P

Steps for calculating miller indices for given directions are:

Find head point coordinate (direction of a head point from the origin (x,y,z) )
Find-tail point coordinate (x,y,z)
Head point coordinate − tail point coordinate

Miller indices for plane A:

Head points = 0,1,1

Tail point = 1/2, 1, 0

Head point − Tail point:

= (−1/2, 0, 1)= (−1/2, 0, 1)×2= (−1, 0, 2)= [1¯ 0 2]

Four-digit system

Convert the three-digitsystem into a four-digit system.

[h,k,l] = [1¯ 0 2]

Coordinates of four-digit system.

h=1/3[2h−k] =1/3[−2−0] =−2/3k=1/3[2k−h] =1/3[0−( −1)] =1/3i=−1/3[h+k] =1/3[−1+0] =+1/3l=−1/3[h+k] =1/3[−1+0] =+1/3

l=l=2[h k i l] = [−2/3, 1/3, +1/3, 2] = [−2/3, 1/3, +1/3, 2]×3 = [−2, 1, +1, 6] = [ 2 ¯ 1 1 6]

Miller indices for plane B:

Head points = 1,0,0

Tail point = 1, 1, 1

Head point − Tail point:

=0, −1, −1= [0 1¯ 1¯]

Four-digit system

Convert the three-digitsystem into a four-digit system.

[h,k,l] = [0 1¯ 1¯]

Coordinates of the four-digit system.

h=1/3[2h−k] =1/3[0−( −1)] =1/3k=1/3[2k−h] =1/3[−2−0] =−2/3i=−1/3[h+k] =1/3[0+( −1)] =1/3l=l=−1[h k i l] = [1/3, −2/3, 1/3, −1 = [1/3, −2/3, 1/3, −1]*3 = [1, −2, 1, −3] = [1 2 ¯ 1 3 ¯]

Miller indices for plane C:

Three digit system:

Head points = 0,0,0

Tail point = 1, -1, 1

Head point − Tail point:

= (−1, 1 −1)= [1¯ 1 1¯]

Four-digit system

Convert three systems into four-digit system.

[h,k,l] = [−1,1,−1]h=1/3[2h−k] =1/3[−2−1] =−1k=1/3[2k−h] =1/3[2+1] =1i=−1/3[h+k] =−1/3[−1+1] =0l=l=−1[h k i l] = [−1, 1, 0, −1] = [ 1 ¯ 1 0 1 ¯]

Conclusion

Miller indices of hexagonal lattice at given directions.

PlaneThree digit systemFour−digit systemA[ 1 ¯ 0 2][ 2 ¯ 1 1 6]B[0 1 ¯ 1 ¯][1 2 ¯ 1 3 ¯]C[ 1 ¯ 1 1 ¯][ 1 ¯ 1 0 1 ¯]

Answer 6

Chapter 3, Problem 3.57P

Interpretation Introduction

Interpretation:

Miller indices for the directions inthe hexagonal lattice using three- digit and four-digit systems.

Concept introduction:

Miller indices at the planes pass through the three points at edges of the crystal structure. In crystalline material, all the points, planes and directions are specified within the unit cell. The direction is defined as a line between two points.

Expert Solution & Answer

Answer to Problem 3.57P

Miller indices of hexagonal lattice at given directions.

PlaneThree digit systemFour−digit systemA[ 1 ¯ 0 2][ 2 ¯ 1 1 6]B[0 1 ¯ 1 ¯][1 2 ¯ 1 3 ¯]C[ 1 ¯ 1 1 ¯][ 1 ¯ 1 0 1 ¯]

Explanation of Solution

Given:

The following hexagonal lattice:

Essentials Of Materials Science And Engineering, Chapter 3, Problem 3.57P

Steps for calculating miller indices for given directions are:

Find head point coordinate (direction of a head point from the origin (x,y,z) )
Find-tail point coordinate (x,y,z)
Head point coordinate − tail point coordinate

Miller indices for plane A:

Head points = 0,1,1

Tail point = 1/2, 1, 0

Head point − Tail point:

= (−1/2, 0, 1)= (−1/2, 0, 1)×2= (−1, 0, 2)= [1¯ 0 2]

Four-digit system

Convert the three-digitsystem into a four-digit system.

[h,k,l] = [1¯ 0 2]

Coordinates of four-digit system.

h=1/3[2h−k] =1/3[−2−0] =−2/3k=1/3[2k−h] =1/3[0−( −1)] =1/3i=−1/3[h+k] =1/3[−1+0] =+1/3l=−1/3[h+k] =1/3[−1+0] =+1/3

l=l=2[h k i l] = [−2/3, 1/3, +1/3, 2] = [−2/3, 1/3, +1/3, 2]×3 = [−2, 1, +1, 6] = [ 2 ¯ 1 1 6]

Miller indices for plane B:

Head points = 1,0,0

Tail point = 1, 1, 1

Head point − Tail point:

=0, −1, −1= [0 1¯ 1¯]

Four-digit system

Convert the three-digitsystem into a four-digit system.

[h,k,l] = [0 1¯ 1¯]

Coordinates of the four-digit system.

h=1/3[2h−k] =1/3[0−( −1)] =1/3k=1/3[2k−h] =1/3[−2−0] =−2/3i=−1/3[h+k] =1/3[0+( −1)] =1/3l=l=−1[h k i l] = [1/3, −2/3, 1/3, −1 = [1/3, −2/3, 1/3, −1]*3 = [1, −2, 1, −3] = [1 2 ¯ 1 3 ¯]

Miller indices for plane C:

Three digit system:

Head points = 0,0,0

Tail point = 1, -1, 1

Head point − Tail point:

= (−1, 1 −1)= [1¯ 1 1¯]

Four-digit system

Convert three systems into four-digit system.

[h,k,l] = [−1,1,−1]h=1/3[2h−k] =1/3[−2−1] =−1k=1/3[2k−h] =1/3[2+1] =1i=−1/3[h+k] =−1/3[−1+1] =0l=l=−1[h k i l] = [−1, 1, 0, −1] = [ 1 ¯ 1 0 1 ¯]

Conclusion

Miller indices of hexagonal lattice at given directions.

PlaneThree digit systemFour−digit systemA[ 1 ¯ 0 2][ 2 ¯ 1 1 6]B[0 1 ¯ 1 ¯][1 2 ¯ 1 3 ¯]C[ 1 ¯ 1 1 ¯][ 1 ¯ 1 0 1 ¯]

Answer 7

Chapter 3, Problem 3.57P

Interpretation Introduction

Interpretation:

Miller indices for the directions inthe hexagonal lattice using three- digit and four-digit systems.

Concept introduction:

Miller indices at the planes pass through the three points at edges of the crystal structure. In crystalline material, all the points, planes and directions are specified within the unit cell. The direction is defined as a line between two points.

Answer 8

Expert Solution & Answer

Answer to Problem 3.57P

Miller indices of hexagonal lattice at given directions.

PlaneThree digit systemFour−digit systemA[ 1 ¯ 0 2][ 2 ¯ 1 1 6]B[0 1 ¯ 1 ¯][1 2 ¯ 1 3 ¯]C[ 1 ¯ 1 1 ¯][ 1 ¯ 1 0 1 ¯]

Answer 9

Expert Solution & Answer

Answer 10

Expert Solution & Answer

Answer 11

Explanation of Solution

Given:

The following hexagonal lattice:

Essentials Of Materials Science And Engineering, Chapter 3, Problem 3.57P

Steps for calculating miller indices for given directions are:

Find head point coordinate (direction of a head point from the origin (x,y,z) )
Find-tail point coordinate (x,y,z)
Head point coordinate − tail point coordinate

Miller indices for plane A:

Head points = 0,1,1

Tail point = 1/2, 1, 0

Head point − Tail point:

= (−1/2, 0, 1)= (−1/2, 0, 1)×2= (−1, 0, 2)= [1¯ 0 2]

Four-digit system

Convert the three-digitsystem into a four-digit system.

[h,k,l] = [1¯ 0 2]

Coordinates of four-digit system.

h=1/3[2h−k] =1/3[−2−0] =−2/3k=1/3[2k−h] =1/3[0−( −1)] =1/3i=−1/3[h+k] =1/3[−1+0] =+1/3l=−1/3[h+k] =1/3[−1+0] =+1/3

l=l=2[h k i l] = [−2/3, 1/3, +1/3, 2] = [−2/3, 1/3, +1/3, 2]×3 = [−2, 1, +1, 6] = [ 2 ¯ 1 1 6]

Miller indices for plane B:

Head points = 1,0,0

Tail point = 1, 1, 1

Head point − Tail point:

=0, −1, −1= [0 1¯ 1¯]

Four-digit system

Convert the three-digitsystem into a four-digit system.

[h,k,l] = [0 1¯ 1¯]

Coordinates of the four-digit system.

h=1/3[2h−k] =1/3[0−( −1)] =1/3k=1/3[2k−h] =1/3[−2−0] =−2/3i=−1/3[h+k] =1/3[0+( −1)] =1/3l=l=−1[h k i l] = [1/3, −2/3, 1/3, −1 = [1/3, −2/3, 1/3, −1]*3 = [1, −2, 1, −3] = [1 2 ¯ 1 3 ¯]