Engineering Calculus Notes 42

Engineering Calculus Notes 42 - dependent or linearly...

This preview shows page 1. Sign up to view the full content.

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
This is the end of the preview. Sign up to access the rest of the document.

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) −→ ı + −→ + −→ k , 2 −→ ı − 2 −→ + 2 −→ k (h) 2 −→ ı − 4 −→ + 2 −→ k , − −→ ı + 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....
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online