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STARTER. What are the two characteristics of a vector?. Vectors and Vector Addition. 1. Characteristics of Vectors 2. Multiplying a vector by a scalar 3. Adding Vectors Graphically 4. Adding Vectors using Components 5. The vector and Scalar Product. What is a vector?.

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  1. STARTER What are the two characteristics of a vector?

  2. Vectors and Vector Addition 1. Characteristics of Vectors 2. Multiplying a vector by a scalar 3. Adding Vectors Graphically 4. Adding Vectors using Components 5. The vector and Scalar Product

  3. What is a vector? • A vector is a mathematical quantity with two characteristics: • Magnitude: “how much”, size of vector, always a non-negative number, “length” of vector • Direction: orientation in space.

  4. Geometrically, a vector is represented as a ray V The terminal or “head” end The initial or “tail” end

  5. The “length” of the ray represents magnitude and direction of A is the angle the ray makes with the +x axis y magnitude V  +X  = 0o

  6. Two vectors A and B are equal if they have the same magnitude and direction. A B This property allows us to move vectors around on our paper/blackboard without changing their properties.

  7. A = -B says that vectors A and B are anti-parallel.They have same size but the opposite direction. B A = -B also implies B = -A A

  8. Vectors can be added geometrically Find A + B B C A A O O B Vector C is the sum of A + B C = A + B

  9. Vector Addition is CommutativeA + B = B + A B Find A + B A A C A O B O B Vector C is the sum of A + B C = A + B = B + A This is the “parallelogram method” learned in trig.

  10. Add vectors “head to tail” A A B O B D C C S could represent four forces acting upon point 0 -tug-of-war D S = A + B + C + D

  11. Prop. #8: Vector subtraction is defined as:A - B = A + (-B) -B A A Find A - B D = A - B B O O -B

  12. Prop. #9: Let A be a vector and k some scalar number. Then kA is a vector with magnitude |k|A. The sign of k dictates the direction of kA. 2A A -3A

  13. A vector A in the x-y plane can be represented by its perpendicular components A y axis Components AX and AY can be positive, negative, or zero. The quadrant that vector A lies in dictates the sign of the components. Components are scalars. AY X axis AX

  14. When the magnitude of vector A is given and its direction specified then its componentscan be computed easily y axis A Ax= Acos AY Ay= Asin  AX X axis

  15. AX is negative whileAY is still positive AX = Acos A y axis AY = Asin AY  The quadrant that A lies in dictates the sign of the trig functions. AX X axis

  16. The magnitude and direction of a vector can be found by knowing its components A = y axis A AY q = tan-1(Ay/Ax) + C  X axis AX

  17. Two Ways to Represent a Vector Magnitude and direction A , q Components Ax, Ay A = q = tan-1(Ay/Ax) + C Ax = A cosq Ay = A sinq

  18. The Scalar or Dot Product Example: A = 3i +4j +5k B = 4i + 2j– 3k A.B = 12 + 8 – 15 = 5

  19. The Cross Product A x B = i( AyBz- AzBy ) - j( AxBz- AzBx ) + k( Ax By- AyBx )

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