Q25E a.聽Consider a vector聽v鈯� in R... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 5: Q25E (page 217)

a.Consider a vector $\overset{鈯�}{v}$ in $R^{n}$ , and a scalar k. Show that $| | k \overset{鈫�}{v} | | = | k | . | | \overset{鈫�}{v} | |$
b.Show that if $\overset{鈯�}{v}$ is a nonzero vector in $R^{n}$ , then
$\overset{鈫�}{u} = \frac{1}{| | \overset{鈫�}{v} | |}$ is a unit vector.

Short Answer

Expert verified

(a) It is proved that, $||k \overset{鈯�}{v} |||k| .|| \overset{鈯�}{v}||$ .

(b) It is proved that $||\overset{鈯�}{u}|| = 1$ .

Step by step solution

Consider for part (a)

Here we want to prove that given any $\overset{鈯�}{v} 鈭� R^{n}$ and for any scalar k, we have $||k \overset{鈫�}{v}|| = |k| . ||\overset{鈯�}{v}||$ .

Observe the following

$鈭� \overset{r}{v} 鈭� = \sqrt{v_{1}^{2} + v_{2}^{2} + 鈥� 鈥� + v_{n}^{2}} Where \overset{r}{V} = (v_{1}, v_{2}, 鈥�, v_{n})$

So,

$\begin{aligned} 鈭� k v 鈭� = \sqrt{{(k v_{1})}^{2} + {(k v_{2})}^{2} + 鈥� . + {(k v_{n})}^{2}} \\ = | k | \sqrt{v_{1}^{2} + v_{2}^{2} + 鈥� . . + v_{n}^{2}} \\ = | k | 鈰� 鈭� \overset{r}{r} 鈭� \end{aligned}$

Consider for part (b).

We will use the part (a) here to prove that $||\overset{鈯�}{u}|| = 1$ .

Observe the following equations.

localid="1659450995384" $\begin{aligned} = |\begin{array}{c} 1 \\ \underset{r}{v} \end{array}| 鈭� \overset{鈫�}{r} v 鈭� \\ = \frac{1}{鈭� v 鈭�} 鈰� 鈭� \overset{r}{v} 鈭� \\ = 1 \end{aligned}$

Hence,

a) $||k \overset{鈯�}{v}|| = |k| . ||\overset{鈯�}{v}||$

b) If V is a non-zero vector in $R^{n}$ then, $\overset{r}{u} = \frac{1}{||\overset{r}{v}||}$ is a unit vector and $||\overset{鈯�}{u}|| = 1$ .

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Short Answer

Step by step solution

Consider for part (a)

Consider for part (b).

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91影视

a.Consider a vectorv鈯� in Rn, and a scalar k. Show that||kv鈫�||=|k|.||v鈫�||b.Show that ifv鈯� is a nonzero vector in Rn, thenu鈫�=1||v鈫�||is a unit vector.

Short Answer

Step by step solution

Consider for part (a)

Consider for part (b).

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Theoretical and Mathematical Physics

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Mechanics Maths

Calculus

Probability and Statistics

Study anywhere. Anytime. Across all devices.