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Vectors

Vectors. An essential component of your physics toolbox. Vectors vs. Scalars. Vector: physical quantity that has both magnitude and direction Examples: displacement, velocity, acceleration Scalar: physical quantity that has magnitude but no direction Examples: distance, speed, time, mass.

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Vectors

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  1. Vectors An essential component of your physics toolbox.

  2. Vectors vs. Scalars • Vector: physical quantity that has both magnitude and direction • Examples: displacement, velocity, acceleration • Scalar: physical quantity that has magnitude but no direction • Examples: distance, speed, time, mass

  3. Vectors vs. Scalars

  4. Representing Vectors • Symbols in text • boldface • Symbols on paper/board • arrow above symbol • Graphically • draw an arrow • length represents magnitude • angle and head of arrow shows direction (angle) • label vector with quantity and/or value v1 = 3 m/s

  5. Adding and Subtracting Vectors • Only vectors of the same quantity and units may be added or subtracted • You can’t add a velocity vector and a displacement vector • You can’t add a velocity in m/s with a velocity in mph • Resultant: vector that represents the sum of two or more vectors

  6. Graphical Addition of Vectors • Draw all vectors to scale. • Add vectors using Head-To-Tail Method • Draw 1st vector • Begin tail of 2nd vector at head of 1st vector • Lather, rinse, repeat • Draw resultant vector from tail of 1st vector to head of last vector. Label resultant.

  7. Resultant of Collinear Vectors

  8. Resultant of Perpendicular Vectors

  9. Properties of Vector Addition • Vectors can be moved parallel to themselves in a diagram. • Useful for switching from Head-To-Tail Method to Parallelogram Method • Vectors can be added in any order • Resultant will always be the same. • To subtract a vector, add its opposite. • Multiplying or dividing vectors by scalars results in vectors. • velocity (vector) = displacement (vector) / time (scalar) • accel (vector) = change in velocity (vector) / time (scalar)

  10. Vectors Can Be Added In Any Order

  11. Vectors Can Be Added In Any Order

  12. Vectors Can Be Added In Any Order

  13. Vector Components:Graphical Representation • projection of vector along x & y axes • how much in the x direction, how much in the y direction • drop projection lines to x & y axes • draw components along axes

  14. Resultant of Two Non-90 Vectors • USE THE COMPONENT METHOD • Add x components • Add y components

  15. Trigonometry Primer • Trigonometric functions relate measurements in right triangles. • Much of our work with vectors can be visualized with right triangles, so we need to know how to do trigonometry.

  16. Trig Primer: Pythagorean Thm. • The Pythagorean Theorem relates the lengths of the sides of a right triangle. c b a

  17. Trig Primer: SOH CAH TOA hyp opp θ adj

  18. Vector Resolution:Finding Vector Components • Trigonometric Method • Visualize vector and its components as a right triangle • Apply Pythagorean Theorem & SOHCAHTOA to find components (“missing sides”)

  19. Vector Addition:Finding a Resultant Vector • Magnitude: • Find the sum of all x-components and all y-components • Use Pythagorean Theorem to find resultant from components.

  20. Vector Addition:Finding a Resultant Vector • Direction: • Apply inverse trig functions to find angle

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