Examples

$(5, 7, 0)$ , $(5, 10, 6)$

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.

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

Step 2

Replace $x_{1}$ , $x_{2}$ , $y_{1}$ , $y_{2}$ , $z_{1}$ , and $z_{2}$ with the corresponding values.

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

Step 3

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

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Step 3.1

Subtract $5$ from $5$ .

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

Step 3.2

Raising $0$ to any positive power yields $0$ .

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

Step 3.3

Subtract $7$ from $10$ .

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

Step 3.4

Raise $3$ to the power of $2$ .

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

Step 3.5

Add $6$ and $0$ .

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

Step 3.6

Raise $6$ to the power of $2$ .

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

Step 3.7

Add $0$ and $9$ .

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

Step 3.8

Add $9$ and $36$ .

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

Step 3.9

Rewrite $45$ as $3^{2} \cdot 5$ .

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Step 3.9.1

Factor $9$ out of $45$ .

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

Step 3.9.2

Rewrite $9$ as $3^{2}$ .

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

Step 3.10

Pull terms out from under the radical.

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

Step 4

The distance between $(5, 7, 0)$ and $(5, 10, 6)$ is $3 \sqrt{5}$ .

$3 \sqrt{5} \approx 6.70820393$

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