Vectors Definitions

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Vectors Definitions
2014-01-19 14:59:32
Vectors Definitions

Vectors Definitions
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  1. Scalar:
    A Quantity that can be expressed by a single number Ex. mass, speed, volume, distance, temperature, work.
  2. Vector:
    A quantity that has direction and magnitude. Ex. velocity, force, displacement, Torque
  3. Opposite Vectors
    Vectors that that have the same magnitude, opposite direction. AB=-BA
  4. Equal Vectors
    Vectors that have the same magnitude and the same direction. AB=AB
  5. Parallel Vectors
    Vectors that have different magnitudes, and same or opposite direction. AB=kAB
  6. Direction of a Vector
    The angle a vector makes when it is placed tail-to-tail with another vector.
  7. Unit Vector
    A vector with a magnitude of 1. It is equal to a vector divided by its magnitude.
  8. Zero Vector
    A vector that has a magnitude of zero and direction undefined. It looks like a point.
  9. Resultant
    The sum of 2 vectors (w=u+v)
  10. Communtativity
    The ability to switch the positions of 2 vectors in vector addition and the dot product.
  11. Associativity
    The ability to group vectors together when doing vector operations.
  12. Distributativity
    The ability to distribute vectors when doing scalar multiplication.
  13. Triangle Inequality
    • The rule that the magnitude of the resultant vector cannot be greater than the sum of the magnitudes of the 2 other vectors.
    • lu+vl<_lul + lvl
  14. State of Equilibrium
    When the resultant of vectors is the zero vector (3 vectors make a triangle)
  15. Force
    The capacity to do work or cause physical change. F=ma
  16. Gravity
    The force that attracts a body towards the centre of the earth or any other physical body that has mass. G=mg (g=9.8m/s2)
  17. Position diagram
    Shows the position of vectors acting on an object (directly from the question)
  18. Vector diagram
    Shows the vector operation that determines the resultant (after manipulating the vectors)
  19. Force parallel
    The force parallel to the ramp which causes the object to slide down. (Fp+Fn=G)
  20. Force Normal
    The force perpendicular to the force parallel that pushed towards the ramp.
  21. Wind Velocity
    The velocity of the wind pushing in a certain direction. Vw+Vg=Va
  22. Ground Velocity
    The velocity of the plane when it is near the ground (without wind)
  23. Air Velocity
    The velocity of the plane in the air. The resultant of the wind velocity and the ground velocity.
  24. Geometric Vectors
    Directed line segment.
  25. Algebraic Vectors
    Directed line segment represented with components. v=(a,b)
  26. One Dimensional Vectors
    Vectors on a line.
  27. R Two Space
    A vector with 2 dimensions (x and y)
  28. R Three Space
    A vector with 3 dimensions (x,y, and z)
  29. Position Vector
    A vector with its tail in the origin.
  30. i^
    A unit vector parallel to the x axis.
  31. j^
    A unit vector parallel to y axis.
  32. k^
    A unit vector parallel to the z axis.
  33. Unit Vector notation
    OP=ai^+bj^+ck^  - A vector represented as the vector sum of scalar multiples of its unit vectors.
  34. Ordered pair/Ordered triple notation
    OP=(a,b,c) - Vector represented with its components.
  35. Direction cosines
    The formulas to find the angles between the vector v and the x,y, and z axis (in R three space)
  36. Direction of a vector in an R Two Space
    The angle between the vector and the positive x axis.
  37. Components
    The coordinates of the endpoint of a position vector.
  38. The Dot product
    The dot product between two vectors is defined by the product of their magnitudes and the cosine angle between the two vectors. It is a scalar value.
  39. The Cross product
    The cross product (x) between the vectors a and b is a vector represented by axb with the following properties:

    • 1. axb is perpendicular to a and b
    • 2. The magnitude of axb is lallblsin@
    • 3. a, b, and axb form a right handed system
  40. Vector projection
    The vector projection (a onto b) is the "shadow" a imposes on b. It is formed by dropping a perpendicular from the tip of a onto b.
  41. Parallelpiped
    A rectangular prism tilted on an angle.
  42. Torque (Nm)
    The turning effect of a force causing an angular effect. (Vector)
  43. Work (J)
    When a force acts on an object and displaces it. (Scalar)
  44. Coplanar
    Vectors are coplanar when they are in the same plane.
  45. Skew lines
    2 lines in an R three space that are not parallel but they never intersect.
  46. Echelon Form
    A type of matrix when the leading entries go down and two the right. Also all the entries below the leading entries are zero and all all zero entries are at the bottom.
  47. Gauss elimination
    The process of transforming an augmented matrix into Echelon form.
  48. Augmented matrix
    A matrix that forms an equation (dotted line represents and = sign)
  49. Row reduced Echelon form
    A type of matrix when:

    • 1. The leading entries go down and to the right
    • 2. The leading entries are all ones
    • 3. There are only zeros above and below the leading entries
    • 4. All all zero rows are at the bottom
  50. Jordan-Gauss Elimination
    The process of transforming an augmented matrix into row reduced Echelon form.