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Chapter 3, Vectors

Chapter 3, Vectors. Outline. Two Dimensional Vectors Magnitude Direction Vector Operations Equality of vectors Vector addition Scalar product of two vectors Vector product of two vectors Multiplication of vectors with scalars. Vectors. General discussion .

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Chapter 3, Vectors

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  1. Chapter 3, Vectors

  2. Outline • Two Dimensional Vectors • Magnitude • Direction • Vector Operations • Equality of vectors • Vector addition • Scalar product of two vectors • Vector product of two vectors • Multiplication of vectors with scalars

  3. Vectors • General discussion. Vector A quantity with magnitude & direction. Scalar A quantity with magnitude only. • Here: We mainly deal with Displacement  D & Velocity  v Discussion is valid for any vector! • Chapter is mostly math! Requires detailed knowledge of trigonometry. • Problem Solving A diagram or sketch is helpful & vital! I don’t see how it is possible to solve a vector problem without a diagram!

  4. Coordinate Systems • Rectangular or Cartesian Coordinates • “Standard” coordinate axes. • Point in the plane is (x,y) • Note, if its convenient could reverse + & -

  5. Plane Polar Coordinates (need trig for understanding!) • Point in the plane is (r,θ)(r = distance from origin, θ= angle from x-axis to line from origin to the point). (a) (b)

  6. Vector & Scalar Quantities Vector  A quantity with magnitude & direction. Scalar  A quantity with magnitude only.

  7. Equality of two vectors • 2 vectors, A & B. A = B means A & B have the same magnitude & direction.

  8. Vector Addition, Graphical Method • Addition of scalars: “Normal” arithmetic! • Addition of vectors: Not so simple! • Vectors in the same direction: • Can also use simple arithmetic Example:Travel 8 km East on day 1, 6 km East on day 2. Displacement = 8 km + 6 km = 14 km East Example:Travel 8 km East on day 1, 6 km West on day 2. Displacement = 8 km - 6 km = 2 km East “Resultant” = Displacement

  9. Adding vectors in same direction:

  10. Graphical Method • For 2 vectors NOTalong same line, adding is more complicated: Example: D1 = 10 km East, D2 = 5 km North. What is the resultant (final) displacement? • 2 methods of vector addition: • Graphical (2 methods of this also!) • Analytical (TRIGONOMETRY)

  11. Graphical Method • 2 vectors NOT along the same line: D1 = 10 km E, D2 = 5 km N. Resultant = ?

  12. Example illustrates general rules (“tail-to-tip” method of graphical addition). Consider V = V1 + V2 1. Draw V1& V2to scale. 2. Place tail of V2at tip of V1 3. Draw arrow from tail of V1to tip of V2 This arrow is the resultantV(measure length and the angle it makes with the x-axis)

  13. Order is not important! V = V1 + V2 = V2 + V1 • In the example, DR = D1 + D2 = D2 + D1 (same as before!)

  14. Graphical Method • Adding (3 or more) vectors V = V1 + V2 + V3

  15. Graphical Method • Second graphical method of adding vectors (equivalent to the tail-to-tip method!) V = V1 + V2 1. Draw V1& V2to scale from common origin. 2. Construct parallelogram using V1& V2 as 2 of the 4 sides. Resultant V = diagonal of parallelogram from common origin(measure length and the angle it makes with the x-axis)

  16. Parallelogram Method

  17. Subtraction of Vectors • First, define the negative of a vector: - V  vector with the same magnitude (size) as V but with opposite direction. Math: V + (- V)  0 • For 2 vectors, A & B: A - B  A + (-B)

  18. Subtraction of 2 Vectors

  19. Multiplication by a Scalar • A vector V can be multiplied by a scalar C V' = C V V'  vector with magnitude CV & same direction as V

  20. Example 3.2 • A two part car trip. First, displacement A = 20 kmdue North. Then, displacement B = 35 km 60º West of North. Figure. Find (graphically) resultant displacement vector R(magnitude & direction). R = A + B Use ruler & protractor to find length of R, angle β. Length = 48.2 km β = 38.9º

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