1 / 27

Starter

Starter. If you are in a large field, what two pieces of information are required for you to locate an object in that field?. Practice – Notes on Vectors. Vectors and Vector Addition. 1. Characteristics of Vectors 2. Multiplying a vector by a scalar 3. Adding Vectors Graphically

cyma
Download Presentation

Starter

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. Starter If you are in a large field, what two pieces of information are required for you to locate an object in that field? Practice – Notes on Vectors

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

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

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

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

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

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

  8. 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. Polar angles are always positive. They go from 0 to 360 degrees.

  9. 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.

  10. 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

  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

  12. 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

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

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

  15. Vector Addition by Components(Do the math)

  16. 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

  17. 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.

  18. 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)

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

  20. 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

  21. 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

  22. 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

  23. 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.

  24. 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

  25. 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

  26. CONNECTION What application of vectors have you seen in real life situations?

  27. Exit: Copy this slide into your notebook 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

More Related