1. © 2013 Pearson Education, Inc. 12G Vectors in Space

2. © 2013 Pearson Education, Inc. 12G Each point in coordinate space can be represented by an ordered triple of the form (x, y, z). The system is similar to the coordinate plane but has an additional coordinate based on the z-axis. Notice that the axes form three planes that intersect at the origin.

3. © 2013 Pearson Education, Inc. 12G Some math teachers, text books, or professors don’t use this orientation for the 3-D graph They change all the axes

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5. A point in 3 Space y - axis z - axis x - axis (2, 3, 1) This is the orientation most mathematicians use.

6. © 2013 Pearson Education, Inc. Every position or point in 3-dimensional space is identified by an ordered triple, (x, y, z). Here is one example of plotting points in 3-dimensional space: 12G y z P (3, 4, 2) The point is 3 units in front of the yz plane, 4 points in front of the xz plane and 2 units up from the xy plane. x

7. © 2013 Pearson Education, Inc. One more time… (2, 3, 1) x Y Z

8. © 2013 Pearson Education, Inc. A Line In Space z x y

9. © 2013 Pearson Education, Inc. Ok. So, how do you exactly do this? Know the axes. The axes in three dimensions are x, y, and z. The coordinate system looks like this: 1. x z y

10. © 2013 Pearson Education, Inc. Let’s graph the point (3, 2, 4) 5. x y z Draw dotted lines parallel to the axes to connect the points drawn so there is a projected point of the desired point on the yz and xz planes.

11. © 2013 Pearson Education, Inc. 12G Graph the point in three-dimensional space. A(3, –2, 1) y x z  From the origin, move 3 units forward along the x-axis, 2 units left, and 1 unit up. The position vector of A is O

12. © 2013 Pearson Education, Inc. 12G Position vectors

13. © 2013 Pearson Education, Inc. 12G

14. © 2013 Pearson Education, Inc. Distance Between Two Points in Space The distance between two points in space is given by the formula: 12G

15. © 2013 Pearson Education, Inc. 12G

16. © 2013 Pearson Education, Inc. 12G

17. © 2013 Pearson Education, Inc. 12G

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19. © 2013 Pearson Education, Inc. 12G

20. © 2013 Pearson Education, Inc. Find the distance between the points P(2, 3, 1) and Q(-3,4,2). Example 1: Solution: Plugging into the distance formula: 12G

21. © 2013 Pearson Education, Inc. Find the distance between the points P(2, 3, 1) and Q(-3,4,2). Example 1: Solution: Plugging into the distance formula: 12G

22. © 2013 Pearson Education, Inc. Find the distance between the points P(2, 3, 1) and Q(-3,4,2). Example 1: Solution: Plugging into the distance formula: 12G

23. © 2013 Pearson Education, Inc. Find the distance between the points P(2, 3, 1) and Q(-3,4,2). Example 1: Solution: Plugging into the distance formula: 12G

24. © 2013 Pearson Education, Inc. Find the distance between the points P(2, 3, 1) and Q(-3,4,2). Example 1: Solution: Plugging into the distance formula: 12G

25. © 2013 Pearson Education, Inc. Example 2: 12G Find the position of P relative to Q given: P(2, 3, 1) and Q(-3,4,2).

26. © 2013 Pearson Education, Inc. Example 2: 12G Find the position of P relative to Q given: P(2, 3, 1) and Q(-3,4,2).

27. © 2013 Pearson Education, Inc. Example 2: 12G Find the vector describing P relative to Q given: P(2, 3, 1) and Q(-3,4,2). I start at Q and go to P. So, how much did I go on the x-axis and in what direction? I start at Q and go to P. So, how much did I go on the y-axis and in what direction? I start at Q and go to P. So, how much did I go on the z-axis and in what direction?

28. © 2013 Pearson Education, Inc. Example 2: 12G Find the vector describing P relative to Q given: P(2, 3, 1) and Q(-3,4,2).

29. © 2013 Pearson Education, Inc. Example 2: 12G Find the position of Q relative to P given: P(2, 3, 1) and Q(-3,4,2).

30. © 2013 Pearson Education, Inc. The Midpoint Between Two Points in Space The midpoint between 2points, is given by: Each coordinate in the midpoint is simply the average of the coordinates in P and Q. Example 2: Find the midpoint of the points P(2, 3, 0) and Q(-4,4,2). 12G

31. © 2013 Pearson Education, Inc. Now, it’s your turn…

32. © 2013 Pearson Education, Inc. HOMEWORK page 299(12G) Numbers 1 – 17; Choose 11 page 302(12H) Numbers 1 – 15; Choose 11