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Chapter 10: Q. 60 (page 802)

Prove Theorem 10.11; that is, show that when $v 鈮� 0$ , the scaled vector role="math" localid="1663644167280" $\frac{1}{|v|} v$ is a unit vector with the same direction as $v$ .

Short Answer

Expert verified

It is proven that, $\frac{1}{|v|} v$ is a unit vector with the same direction as $v$ .

Step by step solution

01

Consider a vector

Let us consider that,

$v = i + j + k$

02

Proof

Now calculate the magnitude of vector $v$ ,

$|v| = \sqrt{1^{2} + 1^{2} + 1^{2}} = \sqrt{1 + 1 + 1} = \sqrt{3}$

The scaled vector $\frac{1}{|v|} v$ is,

$\frac{1}{|v|} v = \frac{1}{\sqrt{3}} (i + j + k) = \frac{1}{\sqrt{3}} i + \frac{1}{\sqrt{3}} j + \frac{1}{\sqrt{3}} k$

The magnitude of the scaled vector $\frac{1}{|v|} v$ is computed as,

$|\frac{1}{|v|} v| = \sqrt{{(\frac{1}{\sqrt{3}})}^{2} + {(\frac{1}{\sqrt{3}})}^{2} + {(\frac{1}{\sqrt{3}})}^{2}} = \sqrt{\frac{1}{3} + \frac{1}{3} + \frac{1}{3}} = \sqrt{\frac{3}{3}} = \sqrt{1} = 1$

Since the magnitude of the scaled vector is unity.

Therefore, the scaled vector is a unit vector.

Hence, proved.

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Most popular questions from this chapter

Find a vector in the direction opposite to $<- 4, 5, - 1>$ and with magnitude 3.

Give precise mathematical definitions or descriptions of each of the concepts that follow. Then illustrate the definition with a graph or algebraic example, if possible.

the formal $未 - M, N - 鈭�,$ , and 狈鈥揗 definitions of the limit statements $\lim_{x 鈫� c} f (x) = 鈭�, \lim_{x 鈫� 鈭�} f (x) = L,$ and $\lim_{x 鈫� 鈭�} f (x) = 鈭�$ , respectively

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: The sum formulas in Theorem 4.4 can be applied only to sums whose starting index value is $k = 1$ .

(b) True or False: ${鈭�}_{k = 0}^{n} 鈥� \frac{1}{k + 1} + {鈭�}_{k = 1}^{n} 鈥� k^{2}$ is equal to ${鈭�}_{k = 0}^{n} 鈥� \frac{k^{3} + k^{2} + 1}{k + 1}$ .

(c) True or False: ${鈭�}_{k = 1}^{n} 鈥� \frac{1}{k + 1} + {鈭�}_{k = 0}^{n} 鈥� k^{2}$ is equal to ${鈭�}_{k = 1}^{n} 鈥� \frac{k^{3} + k^{2} + 1}{k + 1}$ .

(d) True or False: $({鈭�}_{k = 1}^{n} 鈥� \frac{1}{k + 1}) ({鈭�}_{k = 1}^{n} 鈥� k^{2})$ is equal to ${鈭�}_{k = 1}^{n} 鈥� \frac{k^{2}}{k + 1}$ .

(e) True or False: ${鈭�}_{k = 0}^{m} 鈥� \sqrt{k} + {鈭�}_{k = m}^{n} 鈥� \sqrt{k}$ is equal to ${鈭�}_{k = 0}^{n} 鈥� \sqrt{k}$ .

(f) True or False: ${鈭�}_{k = 0}^{n} 鈥� a_{k} = 鈭� a_{0} 鈭� a_{n} + {鈭�}_{k = 1}^{n 鈭� 1} 鈥� a_{k}$ .

(g) True or False: ${({鈭�}_{k = 1}^{10} 鈥� a_{k})}^{2} = {鈭�}_{k = 1}^{10} 鈥� a_{k}^{2}$ .

(h) True or False: ${鈭�}_{k = 1}^{n} 鈥� {(e^{x})}^{2} = \frac{e^{x} (e^{x} + 1) (2 e^{x} + 1)}{6}$ .

In Exercises 22鈥�29 compute the indicated quantities when $u = (2, 1, 鈭� 3), v = (4, 0, 1), and w = (鈭� 2, 6, 5)$

Find the area of the parallelogram determined by the vectors u and v.

Use the definition of the derivative to find $f'$ for each function $f$ in Exercises 39-54

$f (x) = \frac{1}{\sqrt{2 x + 1}}$

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