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Appendix C C-77<br />
Exercise 8 We outline an algebraic derivation of the formula u v u <br />
(v). According to the definition of u v, we need to verify that adding u <br />
(v) to v results in u. Give reasons for each step.<br />
v [u (v)] v [(v) u]<br />
[v (v)] u<br />
O u<br />
u<br />
Exercise 9 Here are examples of more vector algebra properties. Draw two<br />
nonzero vectors u and v with different directions and v not equal to the opposite<br />
of u.<br />
(a) Show 2(u v) 2u 2v.<br />
(b) Show (2 3)u 2u 3u.<br />
(c) Show 2(3u) (2 # 3)u.<br />
Parts (a), (b), and (c) are specific examples of the following vector algebra<br />
properties. For any two vectors u and v and any two scalars k and l,<br />
k(u v) ku kv<br />
(k l)u ku lu<br />
k(lu) (kl)u<br />
Exercise 10 For any vector v, show that 3v (3)v. Hint: 3v is the<br />
opposite of 3v, that is, 3v is the unique vector to add to 3v to get the zero vector.<br />
Use the result from Exercise 9(b) to show that 3v [(3)v] O. Then<br />
(3)v must be 3v.<br />
(5)<br />
(6)<br />
(7)
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