Engineering Calculus Notes 42

Engineering Calculus Notes 42 - dependent or linearly...

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30 CHAPTER 1. COORDINATES AND VECTORS Two (nonzero) vectors are linearly dependent if they point in either the same or opposite directions—that is, if we picture them as arrows from a common initial point, then the two heads and the common tail fall on a line (this terminology will be extended in Exercise 6 —but for more than two vectors, the condition is more complicated). Vectors which are not linearly dependent are linearly independent . Exercises for § 1.2 Practice problems: 1. In each part, you are given two vectors, −→ v and −→ w . Find (i) −→ v + −→ w ; (ii) −→ v −→ w ; (iii) 2 −→ v ; (iv) 3 −→ v 2 −→ w ; (v) the length of −→ v , b −→ v b ; (vi) the unit vector −→ u in the direction of −→ v : (a) −→ v = (3 , 4), −→ w = ( 1 , 2) (b) −→ v = (1 , 2 , 2), −→ w = (2 , 1 , 3) (c) −→ v = 2 −→ ı 2 −→ −→ k , −→ w = 3 −→ ı + −→ 2 −→ k 2. In each case below, decide whether the given vectors are linearly
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Unformatted text preview: dependent or linearly independent. (a) (1 , 2), (2 , 4) (b) (1 , 2), (2 , 1) (c) ( 1 , 2), (3 , 6) (d) ( 1 , 2), (2 , 1) (e) (2 , 2 , 6), ( 3 , 3 , 9) (f) ( 1 , 1 , 3), (3 , 3 , 9) (g) + + k , 2 2 + 2 k (h) 2 4 + 2 k , + 2 k Theory problems: 3. (a) We have seen that the commutative property of vector addition can be interpreted via the parallelogram rule (Figure 5.18 ). Give a similar pictorial interpretation of the associative property. (b) Give geometric arguments for the two distributive properties of vector arithmetic....
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