Examples

(5, 7, 0)

$(5, 7, 0)$ ,

(5, 10, 6)

$(5, 10, 6)$

Step 1

To find the distance between two 3d points, square the difference of the x, y, and z points. Then, sum them and take the square root.

\sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2} + {(z_{2} - z_{1})}^{2}}

$\sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2} + {(z_{2} - z_{1})}^{2}}$

Step 2

Replace

x_{1}

$x_{1}$ ,

x_{2}

$x_{2}$ ,

y_{1}

$y_{1}$ ,

y_{2}

$y_{2}$ ,

z_{1}

$z_{1}$ , and

z_{2}

$z_{2}$ with the corresponding values.

D i s t a n c e = \sqrt{{(5 - 5)}^{2} + {(10 - 7)}^{2} + {(6 + 0)}^{2}}

$D i s t a n c e = \sqrt{{(5 - 5)}^{2} + {(10 - 7)}^{2} + {(6 + 0)}^{2}}$

Step 3

Simplify the expression

\sqrt{{(5 - 5)}^{2} + {(10 - 7)}^{2} + {(6 + 0)}^{2}}

$\sqrt{{(5 - 5)}^{2} + {(10 - 7)}^{2} + {(6 + 0)}^{2}}$ .

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Step 3.1

Subtract

5

$5$ from

5

$5$ .

D i s t a n c e = \sqrt{0^{2} + {(10 - 7)}^{2} + {(6 + 0)}^{2}}

$D i s t a n c e = \sqrt{0^{2} + {(10 - 7)}^{2} + {(6 + 0)}^{2}}$

Step 3.2

Raising

0

$0$ to any positive power yields

0

$0$ .

D i s t a n c e = \sqrt{0 + {(10 - 7)}^{2} + {(6 + 0)}^{2}}

$D i s t a n c e = \sqrt{0 + {(10 - 7)}^{2} + {(6 + 0)}^{2}}$

Step 3.3

Subtract

7

$7$ from

10

$10$ .

D i s t a n c e = \sqrt{0 + 3^{2} + {(6 + 0)}^{2}}

$D i s t a n c e = \sqrt{0 + 3^{2} + {(6 + 0)}^{2}}$

Step 3.4

Raise

3

$3$ to the power of

2

$2$ .

D i s t a n c e = \sqrt{0 + 9 + {(6 + 0)}^{2}}

$D i s t a n c e = \sqrt{0 + 9 + {(6 + 0)}^{2}}$

Step 3.5

Add

6

$6$ and

0

$0$ .

D i s t a n c e = \sqrt{0 + 9 + 6^{2}}

$D i s t a n c e = \sqrt{0 + 9 + 6^{2}}$

Step 3.6

Raise

6

$6$ to the power of

2

$2$ .

D i s t a n c e = \sqrt{0 + 9 + 36}

$D i s t a n c e = \sqrt{0 + 9 + 36}$

Step 3.7

Add

0

$0$ and

9

$9$ .

D i s t a n c e = \sqrt{9 + 36}

$D i s t a n c e = \sqrt{9 + 36}$

Step 3.8

Add

9

$9$ and

36

$36$ .

D i s t a n c e = \sqrt{45}

$D i s t a n c e = \sqrt{45}$

Step 3.9

Rewrite

45

$45$ as

3^{2} \cdot 5

$3^{2} \cdot 5$ .

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Step 3.9.1

Factor

9

$9$ out of

45

$45$ .

D i s t a n c e = \sqrt{9 (5)}

$D i s t a n c e = \sqrt{9 (5)}$

Step 3.9.2

Rewrite

9

$9$ as

3^{2}

$3^{2}$ .

D i s t a n c e = \sqrt{3^{2} \cdot 5}

$D i s t a n c e = \sqrt{3^{2} \cdot 5}$

D i s t a n c e = \sqrt{3^{2} \cdot 5}

$D i s t a n c e = \sqrt{3^{2} \cdot 5}$

Step 3.10

Pull terms out from under the radical.

D i s t a n c e = 3 \sqrt{5}

$D i s t a n c e = 3 \sqrt{5}$

D i s t a n c e = 3 \sqrt{5}

$D i s t a n c e = 3 \sqrt{5}$

Step 4

The distance between

(5, 7, 0)

$(5, 7, 0)$ and

(5, 10, 6)

$(5, 10, 6)$ is

3 \sqrt{5}

$3 \sqrt{5}$ .

3 \sqrt{5} \approx 6.70820393

$3 \sqrt{5} \approx 6.70820393$

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