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Resolving Vectors into Components: A Guide to the Vector Collector Method

Learn how to resolve vectors into components using the Vector Collector method. This guide provides a step-by-step approach to sketching vectors, calculating their x and y components, and reassembling the resultant vector using the Pythagorean theorem. We will explore various vector combinations, including both addition and subtraction, while illustrating key calculations graphically. Through practical examples, you'll gain a clear understanding of how to manipulate vectors in two-dimensional space effectively.

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Resolving Vectors into Components: A Guide to the Vector Collector Method

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  1. Resolving Vectors into Components – the “Vector Collector” Before adding the vectors, you should sketch the vectors. R = J + K J = 160.km [E25oN] K = 95.0km [SE] J 25o K 45o

  2. Resolving Vectors into Components – the “Vector Collector” Sketch the vector’s x & y components R = J + K J = 160.km [E25oN] K = 95.0km [SE] opp adj adj opp J Jy 25o Jx K Ky 45o Kx

  3. Resolving Vectors into Components – the “Vector Collector” Solve for the vector’s x & y components R = J + K J = 160.km [E25oN] K = 95.0km [SE] opp adj adj NOTE: sin 45 = cos 45 (Gotta those 45-45-90 triangles!) opp J Jy 25o Jx K Ky 45o Kx

  4. Resolving Vectors into Components – the “Vector Collector” R = J + K J = 160.km [E25oN] K = 95.0km [SE] opp adj adj add all the X’s to find the total X; then do the same for all the Y’s NOTE: sin 45 = cos 45 (Gotta those 45-45-90 triangles!) opp J Jy 25o Jx K Ky 45o Kx

  5. To re-assemble your vector, use pythagorean theorem R2 = Rx2 + Ry2 = (212)2 + (0.4)2red number = 44944 + 0.16 denotes the R = √44944.16 last significant= 212 km [E Ѳ N] figure tanѲ = Ry so Ѳ = tan-1 Ry Rx Rx = tan-1 0.4 212 = 0.1o ••R = 212 km [E 0.1oN] Rx = 212 km [E] Ry = 0.4 km [N] R Rx Ѳ Ry •

  6. Now try this one, R = P + Q P = 11.0 m [N27oE] Q = 15.0 m [E25oN] First step: Sketch your vectors (no peaking til everybody’s given it a try!!)

  7. R = P + Q P = 11.0 m [N27oE] Q = 15.0 m [E25oN] Px Py 27o P Q Qy Now use the Vector Collector to calculate the components of each vector. 25o Qx

  8. R = P + Q P = 11.0 m [N27oE] Q = 15.0 m [E25oN] Px Py P 27o Q Qy 25o Now reassemble your resultant vector using the pythagorean theorem. Qx

  9. R2 = Rx2 + Ry2 = (18.6)2 + (16.14)2 = 345.96 + 260.4996 R = √606.4596 = 24.6 m [E Ѳ N] Ѳ= tan-1 Ry Rx = tan-1 16.14 18.6 = 40.9o or 41 o R = 24.6 m [E41oN] R = P + Q R Ry Ѳ Rx

  10. How would it change if there was subtraction, R = P – Q ?Think “add the negative” R = P +(–Q) P = 11.0 m [N27oE] Q = 15.0 m [E25oN] -Q = 15.0 m [W25oS] “-Q” is the new name For the vector going in the exact opposite direction of “Q”

  11. R = P + (-Q) P = 11.0 m [N27oE] -Q = 15.0 m [W25oS] Px Py 27o P Q Qy -Qx 25o 25o -Qy Qx -Q Because the directions for -Qx & -Qy have flipped, the resultant components change too!

  12. R = P + (-Q) Ry R R2 = Rx2 + Ry2 = (8.6)2+ (3.46)2 = 73.96 + 11.9716 R = √85.9316 = 9.3 m [W ѲN] Ѳ= tan-1 Ry = tan-1 3.46 = 22o Rx 8.6 R = 9.3 m [W22oN] Rx

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