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Chapter 10: Q 45. (page 801)

Find a vector in the direction of $<8, - 7, 2>$ and with magnitude 2.

Short Answer

Expert verified

The required vector is $\frac{2}{\sqrt{117}} <8, - 7, 2>$ .

Step by step solution

01

Step 1. Given information.

The given vector is:

$v = <8, - 7, 2>$ with magnitude 2.

02

Step 2. Find the norm of the vector.

$v = <8, - 7, 2> ||v|| = \sqrt{8^{2} + {(- 7)}^{2} + 2^{2}} = \sqrt{64 + 49 + 4} = \sqrt{117}$

Therefore, the norm of the given vector is $\sqrt{117}$ .

03

Step 3. Find the vector.

We know that the vector in the direction of vis $\frac{a}{||v||} v$ . Here ais the magnitude of the given vector.

The required vector is:

$\frac{a}{||v||} v = \frac{2}{\sqrt{117}} <8, - 7, 2>$

Therefore the vector in the direction of $v = <8, - 7, 2>$ and with a magnitude of 2 is $\frac{2}{\sqrt{117}} <8, - 7, 2>$ .

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Most popular questions from this chapter

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: $\sum_{k = 0}^{n} \frac{1}{k + 1} + \sum_{k = 1}^{n} k^{2}$ is equal to $\sum_{k = 0}^{n} \frac{k^{3} + k^{2} + 1}{k + 1}$ .

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

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

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

(f) True or False: $\sum_{k = 0}^{n} a_{k} = - a_{0} - a_{n} + \sum_{k = 1}^{n - 1} a_{k}$ .

(g) True or False: ${(\sum_{k = 1}^{10} a_{k})}^{2} = \sum_{k = 1}^{10} a_{k}^{2}$ .

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

Find $||v||$ and find the unit vector in the direction of $v$ .

$v = (3, - 4)$

In Exercises 30–35 compute the indicated quantities when $u = (- 3, 1, - 4), v = (2, 0, 5), and w = (1, 3, 13)$

role="math" localid="1649434688557" $v \times w and w \times v$

In Exercises 24-27, find comp_uv, proj_uv, and the component of v orthogonal tou.

$u = <1, 2>, v = <3, 5>$

If u, v, and ware three mutually orthogonal vectors in $ℝ^{3}$ , explain why $u \times (v \times w) = 0$ .

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