By Alex Bellos

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15. A rower can propel a boat 4 km/hr on a calm river. If the rower rows northwestward against a current of 3 km/hr southward, what is the net velocity of the boat? What is its resultant speed? 16. A singer is walking 3 km/hr southwestward on a moving parade ﬂoat that is being pulled northward at 4 km/hr. What is the net velocity of the singer? What is the singer’s resultant speed? 17. A woman rowing on a wide river wants the resultant (net) velocity of her boat to be 8 km/hr westward. If the current is moving 2 km/hr northeastward,what velocity vector should she maintain?

If x · y Ն 0, then x ϩ y > y . We begin with the desired conclusion x ϩ y > y and try to work “backward” toward the given fact x · y Ն 0, as follows: xϩy > y xϩy > y 2 (x ϩ y) · (x ϩ y) > y 2 x · x ϩ 2x · y ϩ y · y > y 2 x 2 ϩ 2x · yϩ y x 2 2 2 > y 2 ϩ 2x · y > 0. ” However, the last inequality is true if x · y Ն 0. Therefore, we reverse the above steps to create the following “forward” proof of Result 2: Proof. 3 An Introduction to Proof Techniques 33 When “working backward,” your steps must be reversed for the ﬁnal proof.

23. 1. 24. 3. 3. 3. 3. 25. 4. 26. If x is a vector in Rn and c1 ϭ c2 , show that c1 x ϭ c2 x implies that x ϭ 0 (zero vector). 27. True or False: (a) The length of a ϭ [a1 , a2 , a3 ] is a21 ϩ a22 ϩ a23 . (b) For any vectors x, y, z in Rn , (x ϩ y) ϩ z ϭ z ϩ (y ϩ x). (c) [2, 0, Ϫ3] is a linear combination of [1, 0, 0] and [0, 0, 1]. (d) The vectors [3, Ϫ5, 2] and [6, Ϫ10, 5] are parallel. (e) Let x ∈ Rn , and let d be a scalar. If dx ϭ 0, and d ϭ 0, then x ϭ 0. 18 CHAPTER 1 Vectors and Matrices (f) If two nonzero vectors in Rn are parallel, then they are in the same direction.