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Vectors and Vector Addition

Vectors and Vector Addition. 1. Characteristics of Vectors 2. Multiplying a vector by a scalar 3. Adding Vectors Graphically 4. Adding Vectors using Components. What is a vector?. A vector is a mathematical quantity with two characteristics: 1. Magnitude or Length

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Vectors and Vector Addition

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  1. Vectors and Vector Addition 1. Characteristics of Vectors 2. Multiplying a vector by a scalar 3. Adding Vectors Graphically 4. Adding Vectors using Components

  2. What is a vector? A vector is a mathematical quantity with two characteristics: 1. Magnitude or Length 2. Direction ( usually an angle)

  3. Vectors vs. Scalars A vector has a magnitude and direction. Examples: velocity, acceleration, force, torque, etc.

  4. Vectors vs. Scalars A scalar is just a number. Examples: mass, volume, time, temperature, etc.

  5. A vector is represented as a ray,or an arrow. The terminal end or head V The initial end or tail

  6. Picture of a Vector Named A Magnitude of A A = 10 Direction of A q = 30 degrees

  7. The Polar Angle for a Vector Start at the positive x-axis and rotate counter-clockwise until you reach the vector. That’s how you find the polar angle.

  8. Two vectors A and B are equal if they have the same magnitude and direction. A B This property allows us to move vectors around on our paper/blackboard without changing their properties.

  9. A = -B says that vectors A and B are anti-parallel. They have same size but the opposite direction. A = -B also implies B = -A B A

  10. Graphical Addition of Vectors( Head –to Tail Addition ) To find C = A+ B : 1st Put the tail of B on the head of A. 2nd Draw the sum vector with its tail on the tail of A, and its head on the head of B. Example: If C = A+B, draw C. Here’s Vector C

  11. Graphical Addition of Vectors( Head –to Tail Addition ) To find C = A - B : 1st Put the tail of -B on the head of A. 2nd Draw the sum vector with its tail on the tail of A, and its head on the head of -B. Example: If C = A-B, draw C. Here’s Vector C = A - B

  12. Addition of Many Vectors A A B B D C C R D Add A,B,C, and D R= A + B + C + D

  13. Multiplication of a Vector by a Number. A 2A -3A

  14. Vector Addition by Components(Do the math)

  15. A vector A in the x-y plane can be represented by its perpendicular components called Ax and Ay. y Components AX and AY can be positive, negative, or zero. The quadrant that vector A lies in dictates the sign of the components. Components are scalars. A AY x AX

  16. When the magnitude of vector A is given and its direction specified then its componentscan be computed easily y AX= Acosq A AY AY= Asinq  x AX You must use the polar angle in these formulas.

  17. Example: Find the x and y components of the vector shown ifA = 10 and q = 225 degrees. AX = Acosq = 10 cos(225) = -7.07 Ay = Asinq = 10 sin(225) = -7.07 A = (-7.07, -7.07)

  18. The magnitude and polar angle vector can be found by knowing its components A =  = tan-1(AY/AX) + C

  19. Example: Find A, and q if A = ( -7.07, -7.07) = = 10  = tan-1(AY/AX) + C = tan-1(-7.07/-7.07) + 180 = 225 degrees

  20. Example: Find A, and q if A = ( 5.00, -4.00) = = 6.40  = tan-1(AY/AX) + C = tan-1(-4.00/5.00) + 360 = 321 degrees

  21. A vector can be represented by its magnitude and angle, or its x and y components. You can go back and forth’ from each representation with these formulas: If you know Ax and Ay you can get A and q with: If you know A and q, you can get Ax and Ay with: Ax = Acosq Ay = Asinq

  22. Adding Vectors by Components If R = A + B Then Rx = Ax + Bx and Ry = Ay + By So to add vectors, find their components and add the like components.

  23. Example A = ( 3.00,2.00) and B = ( 0, 4.00) If R = A + B find the magnitude and direction of R. Solution: R = A + B = ( 3.00,2.00) + ( 0, 4.00), so R = ( 3.00, 6.00) Then R = ( 32 + 62)1/2 = 6.70 q = tan-1( 6/3) = 63.4o

  24. Example If R = A + B find the magnitude and direction of R. 1st: Find the components of A and B. Ax = 10cos 30 = 8.66 Ay = 10 sin30 = 5.00 Bx = 8cos 135 = -5.66 By = 8sin 135 = 5.66 2nd: Get Rx and Ry Rx = Ax + Bx = 8.66 -5.66 = 3.00 Ry = Ay + By = 5.00 + 5.66 = 10.7 3rd: Get R and q : R = ( 32 + 10.72)1/2 = 11.1 q = tan-1 ( 10.7/3.00) = 74.3o

  25. Summary If you know Ax and Ay you can get A and q with: If you know A and q, you can get Ax and Ay with: Ax = Acosq Ay = Asinq If R = A + B Rx = Ax + Bx Ry = Ay + By

  26. Unit Vectors = ( 1, 0, 0) = ( 0, 1, 0 ) = ( 0, 0, 1) Examples: A = (3,-2,5) = 3i - 2j + 5k B = (3,0,5) = 3i + 5k

  27. The Scalar or Dot Product Example: A = 3i +4j +5k B = 4i + 2j– 3k A.B = 12 + 8 – 15 = 5

  28. The Cross Product A x B = i( Ay Bz- Az By ) - j( Ax Bz- Az Bx ) + k( AxBy- Ay Bx )

  29. Example A = (3,0,2) B = (1,1,0) A X B = = i (0 -2) -j(0-2)+k(3-0) = -2i +2j +3k = (-2,2,3)

  30. Integrals of Vectors dx + j Example: If A = 3xi +5x2j , then find + j

  31. Derivatives of Vectors dA/dx = d (Ax, Ay, Az)/dx = (dAx/dx, dAy/dy, dAz/dz) Example: If A = 3ti - 5t2j , then find dA/dt. dA/dt = 3i -10tj

  32. EXIT If A = 6cos(3t)i +5cos(10t)j , then find

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