Examples

(- 1, - 3, 6)

$(- 1, - 3, 6)$ ,

(- 1, 6, - 4)

$(- 1, 6, - 4)$

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}^{2} + {(y_{2} - y_{1})}_{2}^{2} + {(z_{2} - z_{1})}_{2}^{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{{(- 1 - (- 1))}^{2} + {(6 - (- 3))}^{2} + {(- 4 - 6)}^{2}}

$D i s t a n c e = \sqrt{{(- 1 - (- 1))}^{2} + {(6 - (- 3))}^{2} + {(- 4 - 6)}^{2}}$

Step 3

Simplify the expression

\sqrt{{(- 1 - (- 1))}^{2} + {(6 - (- 3))}^{2} + {(- 4 - 6)}^{2}}

$\sqrt{{(- 1 - (- 1))}^{2} + {(6 - (- 3))}^{2} + {(- 4 - 6)}^{2}}$ .

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply

- 1

$- 1$ by each element of the matrix.

D i s t a n c e = \sqrt{{(- 1 + 1)}^{2} + {(6 - (- 3))}^{2} + {(- 4 - 6)}^{2}}

$D i s t a n c e = \sqrt{{(- 1 + 1)}^{2} + {(6 - (- 3))}^{2} + {(- 4 - 6)}^{2}}$

Step 3.1.2

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

D i s t a n c e = \sqrt{{(- 1 + 1)}^{2} + {(6 - (- 3))}^{2} + {(- 4 - 6)}^{2}}

$D i s t a n c e = \sqrt{{(- 1 + 1)}^{2} + {(6 - (- 3))}^{2} + {(- 4 - 6)}^{2}}$

D i s t a n c e = \sqrt{{(- 1 + 1)}^{2} + {(6 - (- 3))}^{2} + {(- 4 - 6)}^{2}}

$D i s t a n c e = \sqrt{{(- 1 + 1)}^{2} + {(6 - (- 3))}^{2} + {(- 4 - 6)}^{2}}$

Step 3.2

Simplify the expression.

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Step 3.2.1

Add

- 1

$- 1$ and

1

$1$ .

D i s t a n c e = \sqrt{0^{2} + {(6 - (- 3))}^{2} + {(- 4 - 6)}^{2}}

$D i s t a n c e = \sqrt{0^{2} + {(6 - (- 3))}^{2} + {(- 4 - 6)}^{2}}$

Step 3.2.2

Raising

0

$0$ to any positive power yields

0

$0$ .

D i s t a n c e = \sqrt{0 + {(6 - (- 3))}^{2} + {(- 4 - 6)}^{2}}

$D i s t a n c e = \sqrt{0 + {(6 - (- 3))}^{2} + {(- 4 - 6)}^{2}}$

D i s t a n c e = \sqrt{0 + {(6 - (- 3))}^{2} + {(- 4 - 6)}^{2}}

$D i s t a n c e = \sqrt{0 + {(6 - (- 3))}^{2} + {(- 4 - 6)}^{2}}$

Step 3.3

Simplify each term.

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Step 3.3.1

Multiply

- 1

$- 1$ by each element of the matrix.

D i s t a n c e = \sqrt{0 + {(6 + 3)}^{2} + {(- 4 - 6)}^{2}}

$D i s t a n c e = \sqrt{0 + {(6 + 3)}^{2} + {(- 4 - 6)}^{2}}$

Step 3.3.2

Multiply

- 1

$- 1$ by

- 3

$- 3$ .

D i s t a n c e = \sqrt{0 + {(6 + 3)}^{2} + {(- 4 - 6)}^{2}}

$D i s t a n c e = \sqrt{0 + {(6 + 3)}^{2} + {(- 4 - 6)}^{2}}$

D i s t a n c e = \sqrt{0 + {(6 + 3)}^{2} + {(- 4 - 6)}^{2}}

$D i s t a n c e = \sqrt{0 + {(6 + 3)}^{2} + {(- 4 - 6)}^{2}}$

Step 3.4

Simplify the expression.

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Step 3.4.1

Add

6

$6$ and

3

$3$ .

D i s t a n c e = \sqrt{0 + 9^{2} + {(- 4 - 6)}^{2}}

$D i s t a n c e = \sqrt{0 + 9^{2} + {(- 4 - 6)}^{2}}$

Step 3.4.2

Raise

9

$9$ to the power of

2

$2$ .

D i s t a n c e = \sqrt{0 + 81 + {(- 4 - 6)}^{2}}

$D i s t a n c e = \sqrt{0 + 81 + {(- 4 - 6)}^{2}}$

Step 3.4.3

Subtract

6

$6$ from

- 4

$- 4$ .

D i s t a n c e = \sqrt{0 + 81 + {(- 10)}^{2}}

$D i s t a n c e = \sqrt{0 + 81 + {(- 10)}^{2}}$

Step 3.4.4

Raise

- 10

$- 10$ to the power of

2

$2$ .

D i s t a n c e = \sqrt{0 + 81 + 100}

$D i s t a n c e = \sqrt{0 + 81 + 100}$

Step 3.4.5

Add

0

$0$ and

81

$81$ .

D i s t a n c e = \sqrt{81 + 100}

$D i s t a n c e = \sqrt{81 + 100}$

Step 3.4.6

Add

81

$81$ and

100

$100$ .

D i s t a n c e = \sqrt{181}

$D i s t a n c e = \sqrt{181}$

D i s t a n c e = \sqrt{181}

$D i s t a n c e = \sqrt{181}$

D i s t a n c e = \sqrt{181}

$D i s t a n c e = \sqrt{181}$

Step 4

The distance between

(- 1, - 3, 6)

$(- 1, - 3, 6)$ and

(- 1, 6, - 4)

$(- 1, 6, - 4)$ is

\sqrt{181}

$\sqrt{181}$ .

\sqrt{181} \approx 13.45362404

$\sqrt{181} \approx 13.45362404$

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