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Chapter 12 – Vectors and the Geometry of Space

Chapter 12 – Vectors and the Geometry of Space. 12.3 – The Dot Product. Definition – Dot Product. Note: The result is not a vector. It is a real number, a scalar. Sometimes the dot product is called the scalar product or inner product . Example 1 – pg.806 # 8. Find a  b

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Chapter 12 – Vectors and the Geometry of Space

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  1. Chapter 12 – Vectors and the Geometry of Space 12.3 – The Dot Product 12.3 – The Dot Product

  2. Definition – Dot Product Note: The result is not a vector. It is a real number, a scalar. Sometimes the dot product is called the scalar product or inner product. 12.3 – The Dot Product

  3. Example 1 – pg.806 # 8 Find a  b a = 3i + 2j- k b = 4i+ 5k 12.3 – The Dot Product

  4. Properties of the Dot Product 12.3 – The Dot Product

  5. Theorem – Dot Product The dot product can be given a geometric interpretation in terms of the angle  between a and b. 12.3 – The Dot Product

  6. Applying Law of Cosines We can apply the Law of Cosines to the triangle OAB and get the following formulas: 12.3 – The Dot Product

  7. Corollary – Dot Product 12.3 – The Dot Product

  8. Example 2 – pg. 806 # 18 Find the angle between the vectors. (First find an exact expression then approximate to the nearest degree.) a = <4, 0, 2> b = <2, -1, 0> 12.3 – The Dot Product

  9. Orthogonal Vectors Two nonzero a and b are called perpendicular or orthogonal if the angles between them is  = /2. 12.3 – The Dot Product

  10. Hints • The dot product is a way of measuring the extent to which the vectors point in the same direction. • If the dot product is positive, then the vectors point in the same direction. • If the dot product is 0, the vectors are perpendicular. • If the dot product is negative, the vectors point in opposite directions. 12.3 – The Dot Product

  11. Visualization • The Dot Product of Two Vectors 12.3 – The Dot Product

  12. Example 3 For what values of b are the given vectors orthogonal? <-6, b, 2> <b,b2, b> 12.3 – The Dot Product

  13. Definition – Directional Angles The directional angles of a nonzero vector a are the angles , , and  in the interval from 0 to pi that a makes with the positive axes. 12.3 – The Dot Product

  14. Definition – Direction Cosines We get the direction cosines of a vector a by taking the cosines of the direction angles. We get the following formulas 12.3 – The Dot Product

  15. Continued 12.3 – The Dot Product

  16. Example 4 pg. 806 #35 • Find the direction cosines and direction angles of the vector. Give the direction angles correct to the nearest degree. i – 2j – 3k 12.3 – The Dot Product

  17. Definition - Vector Projection • If S is the foot of the perpendicular from R to the line containing , then the vector with representation is called the vector projection of b onto a and is denoted by projab. (think of it as a shadow of b.) 12.3 – The Dot Product

  18. Definition continued 12.3 – The Dot Product

  19. Visualization • Vector Projections 12.3 – The Dot Product

  20. Definition – Scalar Projection The scalar projection or component of b onto a is defined to be the signed magnitude of the vector projection, which is the number |b|cos, where  is the angle between a and b. This is denoted by compab. 12.3 – The Dot Product

  21. Definition continued 12.3 – The Dot Product

  22. Example 5 – pg807 #42 • Find the scalar and vector projections of b onto a. a = <-2, 3, -6> b = <5, -1, 4> 12.3 – The Dot Product

  23. More Examples The video examples below are from section 12.3 in your textbook. Please watch them on your own time for extra instruction. Each video is about 2 minutes in length. • Example 1 • Example 3 • Example 6 12.3 – The Dot Product

  24. Demonstrations Feel free to explore these demonstrations below. • The Dot Product • Vectors in 3D • Vector Projections 12.3 – The Dot Product

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