1. Vectors in Three Dimensions
11.2
Vectors in Three Dimensions

2. Figure 11.25

3. Figure 11.26

4. Figure 11.27

5. Figure 11.28

6. Figure 11.29

7. Figure 11.30

8. Figure 11.31

9. Figure 11.32

10. Figure 11.33

13. Figure 11.34

14. Figure 11.35

15. Figure 11.36

17. Example
Let u = 2, −4, 1 and v = 3, 0, −1 . Find the components of the following vectors and draw them in ℝ3.
a 𝐮
b. −2𝐮
c. 𝐮+2v

18. Figure 11.37

19. Figure 11.38

20. Figure 11.39

21. Figure 11.40

23. Figure 11.41

24. Example Consider the points P(5, 3, 1) and Q(−7, 8, 1).
Express 𝑃𝑄 in terms of the unit vectors i, j, and k
Find the magnitude of 𝑃𝑄 .
Find the position vector of magnitude 10 in the direction of 𝑃𝑄 .

25. Example
A plane is flying horizontally due north in calm air at 300 miles per hour when it encounters a horizontal crosswind blowing southeast at 40 miles per hour and a downdraft blowing vertically downward at 30 miles per hour. What is the resulting speed of the plane relative to the ground? (Finding direction will be presented in the next section.)

26. Figure 11.42