1 / 32

Vectors

Vectors. Definitions. Scalar – magnitude only Vector – magnitude and direction I am traveling at 65 mph – speed is a scalar. It has magnitude but no direction. I am traveling north at 65 mph – speed is a vector. It has both magnitude and direction. Graphical Representation.

toki
Download Presentation

Vectors

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Vectors

  2. Definitions Scalar – magnitude only Vector – magnitude and direction I am traveling at 65 mph – speed is a scalar. It has magnitude but no direction. I am traveling north at 65 mph – speed is a vector. It has both magnitude and direction.

  3. Graphical Representation The vector V is denoted graphically by an arrow. The length of the Arrow represents the magnitude of the vector. The direction of the arrow represents the direction of the vector. V

  4. Vector Representation A vector may be represented by a letter with an arrow over it, e.g. V A vector may be represented by a letter in bold faced type, e.g. V For ease of typing or word processing, vectors will be represented by the bold faced type.

  5. Components of a Vector In two dimensions, a vector will have an x-component (parallel to the X-axis) and a y- component (parallel to the Y-axis). In vector terms, V = Vx+ Vy

  6. Graphically the vector is broken into its components as follows: Y VyV X Vx V = Vx+ Vy

  7. Vector Addition Vectors may be added graphically by placing the tail of the second vector at the head of the first vector and then drawing a new vector from the origin to the head of the second vector. B C C = A + B A

  8. The components of a vector add up to form the vector itself, i.e. V = Vx+ Vyin 2 dimensions V Vy Vx

  9. Or in three dimensions V = Vx+ Vy+ Vz

  10. Components in 3-d Z V Y VxVz Vy X

  11. When we resolve a vector into its components, e.g. V = Vx+ Vy the magnitude of the two component vectors is given by the relations |Vx| = |V| cosϴ |Vy| = |V| sin ϴ

  12. The Pythagorean theorem then gives a relation between the magnitudes of the x and y components, i.e. |V|2 = |Vx |2 + |Vy |2 in 2-dimensions And |V|2 = |Vx |2 + |Vy |2 + |Vz |2 in 3-d

  13. Use of Unit Vectors i j k It is convenient to define three unit vectors iparallel to the X axis j parallel to the Y axis k parallel to the Z axis And to express the components of the vector in terms of a scalar times the unit vector along that axis. Vx = Vxiwhere Vx = |Vx|

  14. Z k j Y I X

  15. Dot or Scalar Product The dot or scalar product of two vectors A · B Is a scalar quantity. A = Axi + Ayj + Azk B = Bxi + Byj + Bzk A · B = |A||Bcosϴ A · B = AxBx + AyBy + AzBz

  16. Example of Dot Product Consider A = 2i + j – 3k B = -i - 3j + k A·B = (2)(-1) + (1)(-3) + (-3)(1) = -2-3-3 = -8 |A| = [22 + 12 + (-3)2]1/2 = [14]1/2 = 3.74 |B| = [(-1)2 + (-3)2 + (1)2]1/2 = [11]1/2 = 3.32 A·B = (3.74)(3.32) cosϴ = 12.41 cosϴ = - 8 cos ϴ = - 8/12.41 = - 0.645 ϴ = cos-1 (- 0.645) = 130.2⁰

  17. Given a vector A = 2i + 3j – k, we can find a vector C that is normal to A by using the fact that the dot product A·C = 0 if A is normal to C. C = Cxi + Cyj + Czk A·C = 2Cx + 3Cy – Cz = 0 You now have three unknowns and only one equation.

  18. How many equations do you need to solve for three unknowns?

  19. I can solve for three unknowns with only this one equation!

  20. A·C = 2Cx + 3Cy – Cz = 0 Let Cy = 1 2Cx + 3 – Cz = 0 Let Cx = 1 2 + 3 – Cz = 0 Cz = 5 So the vector C = i+ j +5k is normal to A.

  21. The order of the vectors in the dot product does not affect the dot product itself, i.e. A · B = B · A

  22. Cross (Vector) Product The cross product of two vectors produces a third vector which is normal to the first two vectors, i.e. C = A x B So vector C is normal to both A and B.

  23. Calculation of C = A x B If the vectors A and B are A = Axi + Ayj + Azk B = Bxi + Byj + Bzk then i j k C = Ax AyAz Bx By Bz

  24. To evaluate the determinant, it is convenient to write the iand j columns to the right and multiply along each of the diagonals 1, 2, and 3 and add them, then multiply along 4, 5, and 6 and subtract them. 1 2 3 4 5 6 i j k i j C = Ax AyAzAx Ay Bx By BzBxBy

  25. C = AyBzi+ AzBxj + AxByk- AzByi– AxBzj - AyBxk Or C = (AyBz – AzBy)i+ (AzBx– AxBz)j + (AxBy– AyBx)k

  26. Example of cross product Calculate C = A x B where A = i + 2j - 3k B = 2i - 3j + k i j k C = 1 2 -3 2 -3 1

  27. C = (2)(1)i + (-3)(2)j + (1)(-3)k - (-3)(-3)i– (1)(1)j – (2)(2)k C = (2 – 9)i + (- 6 – 1)j + (- 3 -4)k C = -7i -7j -7k

  28. To check that we have not made any mistakes in calculating the cross product, we can calculate the dot product C·A which should be equal to 0 since vector C is normal to vectors A and B. C·A = (-7)(1) + (-7)(2) + (-7)(-3) = -7 – 14 +21 = 0 So we have not made any mistakes in calculating C.

  29. The cross product is different if the order is reversed, i.e. A x B = C But B x A = - C B x A = - A x B

  30. When we look at the vector C = -7i – 7j – 7k It has the same direction as the vector C’ = - i – j – k But a different magnitude. |C| = 7 |C’|

  31. Unit Vectors To create a unit vector g in the same direction as G, simply divide the vector by its own magnitude, e.g. g = G/|G| If G = 2i + j – 3k Then |G| = [22 + 12 + (-3)2]1/2 = [4 + 1 + 9]1/2 = [14]1/2 = 3.74 g = (2/3.74)i+ (1/3.74)j – (3/3.74)k

  32. Useful Information A · A = |A|2 i · i = 1 j · j = 1 k · k = 1 A x A = 0 i x i = 0 j x j = 0 k x k = 0 i x j = k j x k = i k x i = j j x i = -k k x j = -ii x k = -j

More Related