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Properties of Scalars and Vectors

Properties of Scalars and Vectors. Vectors. A vector contains two pieces of information: specific numerical value specific direction drawn as arrows on diagrams. Scalars. can be described completely by just one numerical piece of information some are only positive quantities. Vectors.

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Properties of Scalars and Vectors

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  1. Properties of Scalars and Vectors

  2. Vectors • A vector contains two pieces of information: • specific numerical value • specific direction • drawn as arrows on diagrams

  3. Scalars • can be described completely by just one numerical piece of information • some are only positive quantities

  4. Vectors • Notation: Vector: A Length of vector: A or |A| A ≡ |A|

  5. Vector Conventions • In the Cartesian plane, the reference direction is the positive x-axis • Positive angles are measured counter-clockwise • Negative angles are measured clockwise

  6. Vector Conventions • Map directions are always referenced to geographic north, at the top of maps • Angles referenced to true north are indicated by a capital “T” in place of the degree symbol • Three digits are used

  7. A B Equal Vectors have the same magnitude and the same direction A = B

  8. C A vector can be transported as long as its magnitude and direction remain unchanged

  9. A B C Displacement Vector C represents the displacement

  10. Operations with Vectors: Geometric Techniques

  11. V -V 2V -V/2 Adding Vectors If we begin with vector V...

  12. V2 V1 R Adding Vectors If the vectors are unequal... R is called the resultant.

  13. V2 V1 V3 R Adding Vectors You can add more than two vectors...

  14. Vector Subtraction The vector expressions A – B and A +(-B) are equivalent.

  15. -B A A B A – B Vector Subtraction Graphically find A – B.

  16. Operations with Vectors: Mathematical Techniques

  17. Similar Triangles Two triangles are similar when the three angles of one triangle have the same measures as the corresponding angles of the other triangle.

  18. Similar Triangles

  19. Right Triangle a triangle containing one right angle

  20. Right Triangle a triangle containing one right angle the acute angles will always add up to 90°

  21. Right Triangle a triangle containing one right angle the hypotenuse is the side opposite the right angle it is usually labeled “c”

  22. Important Facts to Know • The Pythagorean Theorem: a² + b² = c²

  23. a b sin α = sin β = c c Important Facts to Know • The sine ratio (opp/hyp):

  24. b a cosα = cosβ = c c Important Facts to Know • The cosine ratio (adj/hyp):

  25. a b tan α = tan β = b a Important Facts to Know • The tangent ratio (opp/adj):

  26. 7 tan α = 8 What is the measure of α? α = tan-1(7/8) tan α = 0.875 α 41.2°

  27. Vector Components Every vector has two vector components which are perpendicular to each other. The horizontal component is given a subscript of x: Vx The vertical component is given a subscript of y: Vy

  28. Vector Components If you know the reference angle αfor the vector, its components are found by: |Vx| = V cosα |Vy| = V sin α Assign the correct signs!

  29. N Ny 31 63° Nx Example Nx = 31 cos 63° = 14 units Ny = 31 sin 63° = 28 units Why do we use 2 SDs?

  30. N Ny 31 63° Nx Example Nx = 14 units Ny = 28 units Why are both components positive?

  31. Vector Components It is important to indicate the direction of each component. Down (y) and Left (x) are usually negative (Ex. 4-5).

  32. Vector Components It is important to indicate the direction of each component. Sometimes compass directions are used (Ex. 4-6).

  33. Vector Components In three dimensions, there are x-, y-, and z-components. In three dimensions, there are x-, y-, and z-components. In three dimensions, there are x-, y-, and z-components. By convention, the z-axis is vertical; the others are in the horizontal plane. By convention, the z-axis is vertical; the others are in the horizontal plane. By convention, the z-axis is vertical; the others are in the horizontal plane.

  34. Vector Components Two (or more) vectors can be added by adding their components! Two (or more) vectors can be added by adding their components! Two (or more) vectors can be added by adding their components! (1) Find the x- and y-components of each vector and add them (1) Find the x- and y-components of each vector and add them

  35. Vector Components Two (or more) vectors can be added by adding their components! (2) These are the components of the resultant vector

  36. Vector Components Two (or more) vectors can be added by adding their components! (3) The angle of the resultant vector can also be found with this information

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