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Resolving a Vector into Components - Physics Concept

Learn how to break down a vector into its components in this physics lesson. Discover the process of finding the resultant of several vectors and understand the unlimited possibilities of component arrangements. Explore the concepts of magnitude, direction, and trigonometry in resolving vectors into X and Y components.

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Resolving a Vector into Components - Physics Concept

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  1. PHYSICS VECTORS by Richard J. Terwilliger

  2. Any vector can be broken down into several vectors Let’s look at some examples.

  3. Any vector can be broken down into several vectors We’ll break this GREEN vector up into different RED vectors.

  4. Any vector can be broken down into several vectors Here the GREEN vector is broken down into two RED vectors

  5. The two RED vectors are called the of the GREEN vector. COMPONENTS Any vector can be broken down into several vectors COMPONENT COMPONENT

  6. Or, the of the two RED vectors is the GREEN vector. RESULTANT RESULTANT Any vector can be broken down into several vectors

  7. Any vector can be broken down into several vectors The same GREEN vector can be broken up into 2 different RED vectors.

  8. Any vector can be broken down into several vectors Or two different vectors!

  9. There are possibilities! UNLIMITED Any vector can be broken down into several vectors

  10. Any vector can be broken down into several vectors A little later on we’ll show that two components at right angles are very helpful!

  11. Any vector can be broken down into several vectors Here the same GREEN vector is broken up into 3 different component vectors

  12. Again, the of the three RED vectors is the GREEN vector! RESULTANT Any vector can be broken down into several vectors. RESULTANT

  13. And, the GREEN vector can be broken into RED vectors COMPONENT COMPONENT COMPONENT COMPONENT Any vector can be broken down into several vectors.

  14. Any vector can be broken down into several vectors. Let’s see another example.

  15. Any vector can be broken down into several vectors. This BLUE vector is broken down into several ORANGE vectors.

  16. STARTING POINT Any vector can be broken down into several vectors. The tail of the BLUE vector starts at the same point as the tail of the first ORANGE vector.

  17. STARTING POINT ENDING POINT Any vector can be broken down into several vectors. And they both end at the same point

  18. Any vector can be broken down into several vectors. The of the ORANGE vectors is the BLUEvector. RESULTANT E

  19. Any vector can be broken down into several vectors. Or again, the of the BLUEvectorare the ORANGE vectors. . COMPONENTS E

  20. So what have we learned so far? E

  21. Components are one of two or more vectors having a sum equal to a given vector. A vector can be broken down into any number of components. The different arrangements of components are unlimited. A resultant is the sum of the component vectors. E

  22. The process of finding the components of a vector given it’s magnitude and direction is called: RESOLVING A VECTOR INTO IT'S COMPONENTS E

  23. RESOLVING A VECTOR INTO COMPONENTS is an easy way to find the resultant of several vectors. E

  24. Original Vector (V) COMPONENT 90o COMPONENT RESOLVING a VECTOR into X&Y COMPONENTS To show how, let’s go back to two at right angles. . COMPONENTS E

  25. X-axis TAIL RESOLVING a VECTOR into X&Y COMPONENTS We’ll sketch in the x-axis so it goes through the TAIL of our original vector. Original Vector (V) E

  26. Y-axis X-axis TAIL RESOLVING a VECTOR into X&Y COMPONENTS Then we’ll sketch the y-axis so it goes through the TAIL of our original vector, too. Original Vector (V) E

  27. Y-axis X-axis RESOLVING a VECTOR into X&Y COMPONENTS In the diagram the red vector is called the horizontal or X- component. Original Vector (V) E HORIZONTAL or X-COMPONENT

  28. Y-axis X-axis RESOLVING a VECTOR into X&Y COMPONENTS The blue vector is called the vertical or Y- component. Original Vector (V) VERTICAL or Y-COMPONENT E HORIZONTAL or X-COMPONENT

  29. Y-axis VERTICAL or Y-COMPONENT X-axis RESOLVING a VECTOR into X&Y COMPONENTS To distinguish between the X and Y components, we label each vector with X and Y subscripts respectively. Original Vector (V) E HORIZONTAL or X-COMPONENT

  30. Y-axis VERTICAL or Y-COMPONENT X-axis RESOLVING a VECTOR into X&Y COMPONENTS To distinguish between the X and Y components, we label each vector with X and Y subscripts respectively. Original Vector (V) E Vx

  31. Y-axis X-axis RESOLVING a VECTOR into X&Y COMPONENTS To distinguish between the X and Y components, we label each vector with X and Y subscripts respectively. Original Vector (V) Vy E Vx

  32. RESOLVING a VECTOR into X&Y COMPONENTS Knowing the original vector’s magnitude and direction we can solve for Vx and Vy using trigonometry. Original Vector (V) 90o Vy 180o 0o E Vx 270o

  33. RESOLVING a VECTOR into X&Y COMPONENTS Let’s label the original vector with it’s magnitude and direction. Original Vector (V) 90o Vy 180o 0o E Vx 270o

  34. o 42 RESOLVING a VECTOR into X&Y COMPONENTS Let’s label the original vector with it’s magnitude and direction. V = 36 m/s @ 42 deg 90o Vy 180o 0o E Vx 270o

  35. o 42 RESOLVING a VECTOR into X&Y COMPONENTS To determine the value of the X-component (Vx) we need to use COSINE. V = 36 m/s @ 42 deg 90o Vy 180o 0o E Vx 270o

  36. o 42 RESOLVING a VECTOR into X&Y COMPONENTS Remember COSINE? V = 36 m/s @ 42 deg 90o Vy 180o 0o E Vx 270o

  37. Cosine  = Adjacent Hypotenuse o 42 RESOLVING a VECTOR into X&Y COMPONENTS V = 36 m/s @ 42 deg 90o Vy 180o 0o E Vx 270o

  38. We need to rearrange the equation: solving for the adjacent side. . o 42 Cosine  = Adjacent Hypotenuse RESOLVING a VECTOR into X&Y COMPONENTS V = 36 m/s @ 42 deg 90o Vy 180o 0o E Vx 270o

  39. o 42 Cos  = Adj Hyp RESOLVING a VECTOR into X&Y COMPONENTS therefore Adj=(Hyp)(Cos ) Vx = V cos  V = 36 m/s @ 42 deg 90o Vy 180o 0o E Vx 270o

  40. o 42 Vx = Vcos q Cos  = Adj Hyp RESOLVING a VECTOR into X&Y COMPONENTS therefore Adj=(Hyp)(Cos ) Vx = V cos  V = 36 m/s @ 42 deg 90o Vy 180o 0o E 270o

  41. Vx = V cos  o 42 Vx = Vcos q RESOLVING a VECTOR into X&Y COMPONENTS Vx = 36 m/s (cos 42o) Vx = 26.7 m/s V = 36 m/s @ 42 deg 90o Vy 180o 0o E Vx = 26.7 m/s 270o

  42. o 42 Vx = Vcos q RESOLVING a VECTOR into X&Y COMPONENTS We can now solve for Vy using Sine. V = 36 m/s @ 42 deg 90o Vy 180o 0o E Vx = 26.7 m/s 270o

  43. Sine  = Opposite Hypotenuse o 42 Vx = Vcos q RESOLVING a VECTOR into X&Y COMPONENTS V = 36 m/s @ 42 deg 90o Vy 180o 0o E Vx = 26.7 m/s 270o

  44. o 42 Vx = Vcos q RESOLVING a VECTOR into X&Y COMPONENTS We are solving for the side opposite the 42 degree angle, Vy, therefore we’ll rearrange the equation solving for the opposite side. V = 36 m/s @ 42 deg 90o Vy 180o 0o E Vx = 26.7 m/s 270o

  45. Sine  =Opposite Hypotenuse o 42 Vx = Vcos q RESOLVING a VECTOR into X&Y COMPONENTS Opp = Hyp (Sin ) therefore Vy = V sin  V = 36 m/s @ 42 deg 90o Vy 180o 0o E Vx = 26.7 m/s 270o

  46. Sine  =Opposite Hypotenuse q Vy = Vsin o 42 Vx = Vcos q RESOLVING a VECTOR into X&Y COMPONENTS Opp = Hyp (Sin ) therefore Vy = V sin  V = 36 m/s @ 42 deg 90o 180o 0o E Vx = 26.7 m/s 270o

  47. q Vy = Vsin o 42 Vx = Vcos q RESOLVING a VECTOR into X&Y COMPONENTS Vy = V sin  Vy = 36 m/s (sin 42o) Vy = 24.1 m/s V = 36 m/s @ 42 deg 90o Vy = 24.1 m/s 180o 0o E Vx = 26.7 m/s 270o

  48. q Vy = Vsin o 42 Vx = Vcos q RESOLVING a VECTOR into X&Y COMPONENTS We’ve solved for the horizontal vertical and components of our original vector! V = 36 m/s @ 42 deg 90o Vy = 24.1 m/s 180o 0o E Vx = 26.7 m/s 270o

  49. q Vy = Vsin o 42 Vx = Vcos q RESOLVING a VECTOR into X&Y COMPONENTS Let’s REVIEW what we did. V = 36 m/s @ 42 deg 90o Vy = 24.1 m/s 180o 0o E Vx = 26.7 m/s 270o

  50. V  90o REVIEW RESOLVING a VECTOR into X&Y COMPONENTS A MATHEMATICAL METHOD A vector can be broken down into two vectors at right angles

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