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Chapter 10: Q 43. (page 801)

Find a vector in the direction of $<3, 1, 2>$ and with magnitude 5.

Short Answer

Expert verified

The required vector is $\frac{5}{\sqrt{14}} <3, 1, 2>$ .

Step by step solution

01

Step 1. Given information.

The given vector is:

$v = <3, 1, 2>$ with magnitude 5.

02

Step 2. Find the norm of the vector.

$v = <3, 1, 2> ||v|| = \sqrt{3^{2} + 1^{2} + 2^{2}} = \sqrt{9 + 1 + 4} = \sqrt{14}$

Therefore, the norm of the given vector is $\sqrt{14}$ .

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:

$\frac{a}{||v||} v = \frac{5}{\sqrt{14}} <3, 1, 2>$

Therefore the vector in the direction of $v = <3, 1, 2>$ and with a magnitude of 5 is $\frac{5}{\sqrt{14}} <3, 1, 2>$ .

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Most popular questions from this chapter

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

$u \times v and v \times u$

In Exercises 20-23, find the dot product of the given pairs of vectors and the angle between the two vectors.

$u = <- 5, 1, 3>, v = <- 3, 2, 7>$

what it means, in terms of limits, for a function to have a removable discontinuity, a jump discontinuity, or an infinite discontinuity at x = c

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

In Exercises 24-27, find ${comp}_{u} v, {proj}_{u} v$ and the component of v orthogonal tou.

$u = <3, - 1, 2>, v = <- 4, - 6, 3>$

See all solutions

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