Michael Corral: Vector Calculus

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8 CHAPTER 1. VECTORS IN EUCLIDEAN SPACE Example 1.3. Calculate the following: (a) The magnitude of the vector −→ PQ in 2 with P=(−1,2) and Q=(5,5). Solution: By formula (1.2), ∥ ∥ ∥ −→ PQ ∥ ∥∥= √ (5−(−1)) 2 +(5−2) 2 = √ 36+9= √ 45=3 √ 5. (b) The magnitude of the vector v=(8,3) in 2 . Solution: By formula (1.3),‖v‖= √ 8 2 +3 2 = √ 73. (c) The distance between the points P=(2,−1,4) and Q=(4,2,−3) in 2 . Solution: By formula (1.4), the distance d= √ (4−2) 2 +(2−(−1)) 2 +(−3−4) 2 = √ 4+9+49= √ 62. (d) The magnitude of the vector v=(5,8,−2) in 3 . Solution: By formula (1.5),‖v‖= √ 5 2 +8 2 +(−2) 2 = √ 25+64+4= √ 93. A ☛ ✟ ✡Exercises ✠ 1. Calculate the magnitudes of the following vectors: (a) v=(2,−1) (b) v=(2,−1,0) (c) v=(3,2,−2) (d) v=(0,0,1) (e) v=(6,4,−4) 2. For the points P=(1,−1,1), Q=(2,−2,2), R=(2,0,1), S= (3,−1,2), does −→ PQ= −→ RS? 3. For the points P=(0,0,0), Q=(1,3,2), R=(1,0,1), S= (2,3,4), does −→ PQ= −→ RS? B 4. Let v=(1,0,0) and w=(a,0,0) be vectors in 3 . Show that‖w‖=|a|‖v‖. 5. Let v=(a,b,c) and w=(3a,3b,3c) be vectors in 3 . Show that‖w‖=3‖v‖. C 6. ThoughwewillseeasimpleproofofTheorem1.1 in the next section, it is possible to prove it using methods similar to those in the proof of Theorem 1.2. Prove the special case of Theorem 1.1 where the points P=(x 1 ,y 1 ,z 1 ) and Q=(x 2 ,y 2 ,z 2 ) satisfy the following conditions: x 2 > x 1 > 0, y 2 > y 1 > 0, and z 2 > z 1 > 0. (Hint: Think of Case 4 in the proof of Theorem 1.2, and consider Figure 1.1.9.) x z P(x 1 ,y 1 ,z 1 ) 0 S(x 1 ,y 1 ,0) U(x 2 ,y 1 ,0) Q(x 2 ,y 2 ,z 2 ) R(x 2 ,y 2 ,z 1 ) y T(x 2 ,y 2 ,0) Figure 1.1.9
1.2 <strong>Vector</strong> Algebra 9 1.2 <strong>Vector</strong> Algebra Now that we know what vectors are, we can start to perform some of the usual algebraic operations on them (e.g. addition, subtraction). Before doing that, we will introduce the notion of a scalar. Definition 1.3. A scalar is a quantity that can be represented by a single number. For our purposes, scalars will always be real numbers. 3 Examples of scalar quantities are mass, electric charge, and speed (not velocity). 4 We can now define scalar multiplication of a vector. Definition 1.4. For a scalar k and a nonzero vector v, the scalar multiple of v by k, denoted by kv, is the vector whose magnitude is|k|‖v‖, points in the same direction as v if k>0, points in the opposite direction as v if k
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188 Bibliography Pogorelov, A.V., A
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190 Appendix A: Answers and Hints t
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Appendix B We will prove the right-
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Appendix C 3D Graphing with Gnuplot
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202 GNU Free Documentation License
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Index Symbols D....................
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Michael Corral: Vector Calculus

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