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Fun with Vectors

Fun with Vectors. Definition. A vector is a quantity that has both magnitude and direction Examples?. v, v , or AB. Represented by an arrow. B (terminal point). A (initial point). If two vectors, u and v , have the same length and direction, we say they are equivalent. v. u.

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Fun with Vectors

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  1. Fun with Vectors

  2. Definition • A vector is a quantity that has both magnitude and direction • Examples?

  3. v, v, or AB Represented by an arrow B (terminal point) A (initial point)

  4. If two vectors, u and v, have the same length and direction, we say they are equivalent v u

  5. Vector addition b a

  6. Vector addition: a + b b a

  7. Vector addition: a + b b a+b a

  8. Vector addition: a + b b a+b a

  9. Scalar Multiplication a

  10. Scalar Multiplication 2a a -½ a -a

  11. Subtraction b a

  12. Subtraction b a -b

  13. Subtraction b a a+(-b) -b

  14. Subtraction b a a+(-b) -b

  15. Subtraction If a and b share the same initial point, the vector a-b is the vector from the terminal point of b to the terminal point of a a+(-b) b a -b

  16. Let’s put these on a coordinate system We can describe a vector by putting its initial point at the origin. We denote this as a=<a1,a2> where (a1,a2) represent the terminal point

  17. z y (a,b,c) c y a x b x v=<a,b,c> Graphically (a1,a2) a=<a1,a2>

  18. Given two points A=(x1,y1) and B=(x2,y2), The vector v = AB is given by v = <x2 - x1, y2 - y1> …or in 3-space, v = <x2 - x1, y2 - y1, z2 - z1>

  19. B v A Graphically A=(-1,2) B=(2,3) v = <2-(-1), 3-2> = <3,1>

  20. Recall, a vector has direction and length Definition: The magnitude of a vector v = <x,y,z> is given by

  21. Properties of VectorsSuppose a, band c are vectors, c and d are scalars • a+b=b+a • a+(b+c)=(a+b)+c • a+0=a • a+(-a)=0 • c(a+b)=ca+cb • (c+d)a=ca+da • (cd)a=c(da) • 1a=a

  22. Standard Basis Vectors Definition: vectors with length 1 are called unit vectors

  23. Example: We can express vectors in terms of this basis a = <2,-4,6> a = 2i -4j+6k Q. How do we find a unit vector in the same direction as a? A. Scale a by its magnitude

  24. Example a = <2,-4,6>

  25. 12.3 The Dot Product Motivation: Work = Force* Distance F Fy D Fx Box

  26. F Fy D Fx Box To find the work done in moving the box, we want the part of F in the direction of the distance

  27. One interpretation of the dot product Where is the angle between F and D

  28. z y x A more useful definition You can show these two definitions are equal by considering the following triangle and applying the law of cosines! See page 808 for details a-b b a Think, what is |a|2?

  29. Example a=<2,-1,0>, b=<1,-8,-3> Find a.b and the angle between a and b

  30. The Dot Product If a = <a1,a2,a3> and b=<b1,b2,b3> then The dot product of a and b is a NUMB3R given by

  31. The Dot Product a a and b are orthogonal if and only if the dot product of a and b is 0 b Other Remarks:

  32. Properties of the dot productSuppose a, b, and c are vectors and c is a scalar • a.a=|a|2 • a.b=b.a • a.(b+c) = (a.b)+(a.c) • (ca).b=c(a.b)=a.(cb) • 0.a=0

  33. Yet another use of the dot product: Projections a.b=|a| |b| cos( ) b a |b| cos( ) Think of our work example: this is ‘how much’ of b is in the direction of a

  34. We call this quantity the scalar projection of b on a Think of it this way: The scalar projection is the length of the shadow of b cast upon a by a light directly above a

  35. Q. How do we get the vector in the direction of a with length compab? • We need to multiply the unit • vector in the direction of a by compab. We call this the vector projection of b onto a

  36. Examples/Practice!

  37. Key Points • Vector algebra: addition, subtraction, scalar multiplication • Geometric interpretation • Unit vectors • The dot product and the angle between vectors • Projections (algebraic and geometric)

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