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Chapter 9: Problem 22

Find the distance between the points. \((2,2,3), \quad(4,-5,6)\)

Short Answer

Expert verified
The distance between the points \( (2,2,3) \) and \( (4,-5,6) \) is \( \sqrt{62} \).

Step by step solution

01

Identify Points in Space

The given two points in the three-dimensional space are \( (2,2,3) \) and \( (4,-5,6) \). Let's label them as \( P1(x_1, y_1, z_1) = (2,2,3) \) and \( P2(x_2, y_2, z_2) = (4,-5,6) \).
02

Apply the Euclidean Distance Formula

The Euclidean distance between two points \( P1(x_1, y_1, z_1) \) and \( P2(x_2, y_2, z_2) \) in a three-dimensional space is given by the formula \( D = \sqrt{{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}} \). Let's substitute the coordinates into the distance formula.
03

Substitute the coordinates

Substituting the coordinates, we have \( D = \sqrt{{(4 - 2)^2 + ((-5) - 2)^2 + (6 - 3)^2}} \).
04

Simplify the Expression

After simplifying the above expression, we get \( D = \sqrt{{2^2 + (-7)^2 + 3^2}} = \sqrt{{4 + 49 + 9}} = \sqrt{62} \).
05

Final Answer

So, the distance between the given two points is \( \sqrt{62} \).

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Most popular questions from this chapter

Find the angle \(\theta\) between the vectors. $$ \begin{array}{l} \mathbf{u}=\langle 1,1,1\rangle \\ \mathbf{v}=\langle 2,1,-1\rangle \end{array} $$

In Exercises 69 and \(70,\) find a unit vector \((a)\) in the direction of \(\mathbf{u}\) and \((\mathbf{b})\) in the direction opposite \(\mathbf{u}\) \(\mathbf{u}=\langle 2,-1,2\rangle\)

Determine which of the vectors is (are) parallel to \(\mathrm{z}\). Use a graphing utility to confirm your results. \(\mathbf{z}\) has initial point (5,4,1) and terminal point (-2,-4,4) (a) \langle 7,6,2\rangle (b) \langle 14,16,-6\rangle

Determine which of the vectors is (are) parallel to \(\mathrm{z}\). Use a graphing utility to confirm your results. \(\mathbf{z}=\frac{1}{2} \mathbf{i}-\frac{2}{3} \mathbf{j}+\frac{3}{4} \mathbf{k}\) (a) \(6 \mathbf{i}-4 \mathbf{j}+9 \mathbf{k}\) (b) \(-\mathbf{i}+\frac{4}{3} \mathbf{j}-\frac{3}{2} \mathbf{k}\) (c) \(12 \mathbf{i}+9 \mathbf{k}\) (d) \(\frac{3}{4} \mathbf{i}-\mathbf{j}+\frac{9}{8} \mathbf{k}\)

Find \(u \cdot v\). \(\|\mathbf{u}\|=40,\|\mathbf{v}\|=25,\) and the angle between \(\mathbf{u}\) and \(\mathbf{v}\) is \(5 \pi / 6\).
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