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Chapter 10: Q 3 (page 848)

Find the distance between the points (1, 5,-2) and (3, 9,-1).

Short Answer

Expert verified

The required distance is $\sqrt{21}$ .

Step by step solution

Given information

Given two points in $ℝ^{3}, (1, 5, - 2) a n d (3, 9, - 1)$

Finding the distance

The distance between the given points as per formula, is:

$\sqrt{{(3 - 1)}^{2} + {(9 - 5)}^{2} + {(- 1 - (- 2))}^{2}}$

Simplifying:

$\sqrt{4 + 16 + 1}$

Simplify further:

$\sqrt{21}$

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Most popular questions from this chapter

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

In Exercises 36–41 use the given sets of points to find:

(a) A nonzero vector N perpendicular to the plane determined by the points.

(b) Two unit vectors perpendicular to the plane determined by the points.

$P (4, - 2), Q (- 2, 0), R (1, - 5)$

(Hint: Think of the $x y$ -plane as part of $ℝ^{3}$ .)

Give an example of three vectors in $ℝ^{3}$ that form a right-handed triple. Explain how you can use the same three vectors to form a left-handed triple.

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

What is a parallelepiped? What is meant by the parallelepiped determined by the vectors u, v and w? How do you find the volume of the parallelepiped determined by u, v and w?

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Find the distance between the points (1, 5,-2) and (3, 9,-1).

Short Answer

Step by step solution

Given information

Finding the distance

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