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Vectors 1

Vectors 1. What is a vector?. A vector is a mathematical quantity with two characteristics: 1) magnitude: “how much”, size of vector, always a non-negative number, “length” of vector 2) Direction: orientation in space. Can you think of science quantities which are vectors in nature?

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Vectors 1

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  1. Vectors 1

  2. What is a vector? • A vector is a mathematical quantity with two characteristics: • 1) magnitude: “how much”, size of vector, always a non-negative number, “length” of vector • 2) Direction: orientation in space. • Can you think of science quantities which are vectors in nature? • Students take notes:

  3. Vectors vs. scalar quantities • A scalar is a quantity which has only the characteristic of magnitude. • Name some scalars in science... • Student answers/suggestions:

  4. In art!

  5. In navigation New York Harbor

  6. Prop. #1: Geometrically, a vector is represented as a ray V The terminal or “head” end The initial or “tail” end

  7. Prop. #2: 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

  8. Prop. #3: 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.

  9. Prop. #4: 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

  10. Prop. 5: Vectors can be added geometrically Find A + B B C A A O B O B Vector C is the sum of A + B C = A + B

  11. Prop. #6: 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.

  12. Prop. #7: Add vectors “head to head” 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

  13. Prop. #8: Vector subtraction is defined as:A - B = A + (-B) -B A A Find A - B D = A - B B O B O -B A B -B Note that the magnitude of D could be larger than the magnitude of the sum C. C = A + B D = A - B O B -B

  14. 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

  15. 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

  16. 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

  17. 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

  18. Magnitude and direction of a vector can be found by knowing its components y axis A AY tan = AY/AX  X axis AX  = tan-1(AY/AX)

  19. That is all for this slide file.... for this moment...

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