Michael Corral: Vector Calculus

1.3 Dot Product 19
3. v=(5,1,−2), w=(4,−4,3) 4. v=(7,2,−10), w=(2,6,4)
5. v=(2,1,4), w=(1,−2,0) 6. v=(4,2,−1), w=(8,4,−2)
7. v=−i+2j+k, w=−3i+6j+3k 8. v=i, w=3i+2j+4k
9. Let v=(8,4,3) and w=(−2,1,4). Is v⊥w? Justify your answer.
10. Let v=(6,0,4) and w=(0,2,−1). Is v⊥w? Justify your answer.
11. For v, w from Exercise 5, verify the Cauchy-Schwarz Inequality|v·w|≤‖v‖‖w‖.
12. For v, w from Exercise 6, verify the Cauchy-Schwarz Inequality|v·w|≤‖v‖‖w‖.
13. For v, w from Exercise 5, verify the Triangle Inequality‖v+w‖≤‖v‖+‖w‖.
14. For v, w from Exercise 6, verify the Triangle Inequality‖v+w‖≤‖v‖+‖w‖.
B
Note: Consider only vectors in 3 for Exercises 15-25.
15. Prove Theorem 1.9(a). 16. Prove Theorem 1.9(b).
17. Prove Theorem 1.9(c). 18. Prove Theorem 1.9(d).
19. Prove Theorem 1.9(e). 20. Prove Theorem 1.10(a).
21. Prove or give a counterexample: If u·v=u·w, then v=w.
C
22. Prove or give a counterexample: If v·w=0for all v, then w=0.
23. Prove or give a counterexample: If u·v=u·w for all u, then v=w.
24. Prove that ∣ ∣
∣‖v‖−‖w‖
∣∣≤‖v−w‖
for all v, w.
v
25. For nonzero vectors v and w, the projection of v onto w
(sometimes written as proj w v) is the vector u along the
same line L as w whose terminal point is obtained by dropping
a perpendicular line from the terminal point of v to L w
L
u
(see Figure 1.3.5). Show that
Figure 1.3.5
‖u‖= |v·w|
‖w‖ .
(Hint: Consider the angle between v and w.)
26. Letα,β, andγbe the angles between a nonzero vector v in 3 and the vectors i, j,
and k, respectively. Show that cos 2 α+cos 2 β+cos 2 γ=1.
(Note: α,β,γare often called the direction angles of v, and cosα, cosβ, cosγ are
called the direction cosines.)