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VECTORS

Learn about vectors, scalar quantities, drawing vectors, vector magnitude, equal vectors, inverse vectors, vector properties, addition of vectors, and vector components.

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VECTORS

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  1. VECTORS VECTORS ARE QUANTITIES THAT HAVE MAGNITUDE AND DIRECTION

  2. Vector vs. Scalar • Scalar- Quantities that have magnitude(size)only, like mass, time, distance, energy, volume, and speed. • Vector- Quantities that have both magnitude and direction like displacement, velocity, acceleration, force, momentum, and fields.

  3. DRAWING A VECTOR? A vector has both size and direction. Tip A vector is represented on paper by an arrow drawn to scale and pointing in the direction of the action Tail

  4. Magnitude of Vectors • The best way to describe the magnitude of a vector is to measure the length of the vector. • The length of the vector is proportional to the magnitude of the quantity it represents.

  5. If vector A represents a displacement of three miles to the north… A B Then vector B, which is twice as long, would represent a displacement of six miles to the north! Magnitude of Vectors

  6. Equal Vectors Equal vectors have the same length and direction, and must represent the same quantity (such as force or velocity).

  7. A -A Inverse Vectors Inverse vectors have the same length, but opposite direction.

  8. VECTOR PROPERTIES • Commutative Property: A+B = B+A • Associative Property: (A+B)+C = A+(B+C) • Zero Property: A+(-B) = 0, if and only if, A is equal in magnitude to B and pointing in the opposite direction. • Subtraction: A - B = A + (-B) • Multiplication: 3 x A = 3A

  9. Vector angle ranges NW quad 270o <  < 360o NE quad 0 <  < 90o     SW quad 180o <  < 270o SE quad 90o <  < 180o N E W S

  10. ADDITION OF VECTORS • 3 Methods • Parallelogram Method- For a quick assessment. Good for concurrent forces. • Tip-to-Tail Method- Drawing vectors to scale on paper to find an answer. Use of a pencil, ruler and protractor needed. Good for displacements. • Mathematical Method- Determining an answer using trigonometry. The vectors need to be at right angles to one another.

  11. PARALLELOGRAM METHOD • Arrange the vectors tail to tail in the correct direction and draw to scale. • Draw two identical vectors as the originals to form a parallelogram. • Draw in the diagonal of the parallelogram. This is your answer called a resultant. • Measure the resultant and find the angle.

  12. A R B B A THE PARALLELOGRAM METHOD

  13. Concurrent Forces

  14. The Resultant and the Equilibrant The sum of two or more vectors is called the resultant vector. The resultant vector can replace the vectors from which it is derived. The resultant is completely canceled out by adding it to its inverse, which is called the equilibrant.

  15. TIP-TO-TAIL METHOD • Arrange the scaled vectors from the tip of one to the tail of the next. • Draw the resultant from the tail of the first vector to the tip of the last vector. • Determine the magnitude of the resultant, and find the angle from the base of the resultant. Use a ruler and protractor.

  16. Displacements as Vectors Direction Magnitude

  17. A Scale and Ruler

  18. The Protractor The obtuse angle The acute angle

  19. A: 12 meters 20o East of North C B A R B: 15 meters East Find the Resultant Displacement C: 5 meters 30o North of West TIP-TO-TAIL METHOD

  20. MATHEMATICAL METHODfor vectors at right angles • Sketch a diagram of the vectors. • Use the pythagorean theorem to determine the magnitude of the resultant. • Use the sine, cosine, or tangent function to determine the angle from the base of the resultant.

  21. T.O.A.S.O.H.C.A.H. TRIG REVIEW SOHCAHTOA Pythagorean Theorem hyp opp q adj

  22. MATHEMATICAL METHOD A: 85 N, West B: 150 N, North B R R B Find the Resultant A

  23. VECTOR COMPONENTS • Every vector has 2 components. • One component is horizontal, or x-direction. • One component is vertical, or y-direction. • The 2 components are always  to each other. • Use trig functions to find them.

  24. Any Old Vector Y-Component X-Component

  25. HORIZONTAL COMPONENT VERTICAL COMPONENT

  26. Inclined Planes Ff FN mgcosq q mg mgsinq q

  27. IN SUMMARY • Vectors have magnitude and direction • 3 methods to add vectors • Parallelogram • Tip-to-Tail • Mathematical • Components are perpendicular vectors to any old vector.

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