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Component Vectors

Component Vectors. Vectors have two parts (components) X component – along the x axis Y component – along the y axis. Finding components. X component – follow from the tail of the vector along the x axis until you reach the point where the tip would be if it fell straight down.

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Component Vectors

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  1. Component Vectors • Vectors have two parts (components) • X component – along the x axis • Y component – along the y axis

  2. Finding components • X component – follow from the tail of the vector along the x axis until you reach the point where the tip would be if it fell straight down

  3. Y component – follow from where you stopped on the x axis straight up to the tip • You should now have formed a right triangle with the original vector as the hypotenuse

  4. To find components • To find components, you must use trigonometric functions Hypotenuse Opposite ø Adjacent

  5. Trig functions • Θ is the angle between the vector and the x axis • sin Θ = _opposite_ hypotenuse • cos Θ = _adjacent_ hypotenuse • tan Θ = _opposite_ adjacent

  6. Steps for finding the components • Draw a picture (arrowheads, original vector & components) • Choose a trig function • Use algebra to solve for the desired variable & plug in • Calculator in degrees! • Check with Pythagorean theorem

  7. Example

  8. X component • cos Θ = _adjacent_ hypotenuse • cos 35 = _adjacent_ 316 • 316 cos 35 = adjacent • 259 N = adjacent

  9. Y component • sin Θ = _opposite_ hypotenuse • sin 35 = _opposite_ 316 • 316 sin 35 = opposite • 181 N = opposite

  10. How to find components when you add two vectors • Find the x and y component for both vectors • Add up the x components • Add up the y components • Draw a new set of vectors • Use Pythagorean theorem to get the magnitude of the resultant vector • Use arctangent to get the angle of the new vector

  11. X component adj = hyp cos Θ adj = 36 cos34º adj = + 29.8 m Y component opp = hyp sin Θ opp = 36 sin34º opp = +20.1 m Vector d1

  12. X component opp = hyp sin Ø opp = 23 sin64º opp = - 20.7 m Y component adj = hyp cos Θ adj = 23 cos64º adj = +10.1 m Vector d2

  13. Total X displacement – add d1 and d2 dtotal = d1 + d2 dtotal = 29.8 m + (-20.7m) dtotal = +9.1m

  14. Total Y displacement – add d1 and d2 dtotal = d1 + d2 dtotal = 20.1 m + 10.1m dtotal = +30.2m

  15. To get the magnitude of the resultant vector • Use Pythagorean Theorem dTotal = (dX)2+ (dy)2 dTotal = (9.1)2+ (30.2)2 dTotal = 82.81+ 912.04 dTotal = 994.85 = 31.5 m

  16. To find the angle of the resultant vector • Use arctangent function: Θ = tan-1 (opp/adj) Θ = tan-1 (30.2/9.1) Θ = tan-1 (3.3) Θ = 73.1°

  17. Formulas • a2 + b2 = c2 • R2 = a2 + b2 - 2ab(cosθ) • SOH • CAH • TOA

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