Algebra Examples

(1, 2, 3)

$(1, 2, 3)$ ,

(3, 2, 1)

$(3, 2, 1)$

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{{(3 - 1)}^{2} + {(2 - 2)}^{2} + {(1 - 3)}^{2}}

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

Step 3

Simplify the expression

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

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

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Step 3.1

Subtract

1

$1$ from

3

$3$ .

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

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

Step 3.2

Raise

2

$2$ to the power of

2

$2$ .

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

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

Step 3.3

Subtract

2

$2$ from

2

$2$ .

D i s t a n c e = \sqrt{4 + 0^{2} + {(1 - 3)}^{2}}

$D i s t a n c e = \sqrt{4 + 0^{2} + {(1 - 3)}^{2}}$

Step 3.4

Raising

0

$0$ to any positive power yields

0

$0$ .

D i s t a n c e = \sqrt{4 + 0 + {(1 - 3)}^{2}}

$D i s t a n c e = \sqrt{4 + 0 + {(1 - 3)}^{2}}$

Step 3.5

Subtract

3

$3$ from

1

$1$ .

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

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

Step 3.6

Raise

- 2

$- 2$ to the power of

2

$2$ .

D i s t a n c e = \sqrt{4 + 0 + 4}

$D i s t a n c e = \sqrt{4 + 0 + 4}$

Step 3.7

Add

4

$4$ and

0

$0$ .

D i s t a n c e = \sqrt{4 + 4}

$D i s t a n c e = \sqrt{4 + 4}$

Step 3.8

Add

4

$4$ and

4

$4$ .

D i s t a n c e = \sqrt{8}

$D i s t a n c e = \sqrt{8}$

Step 3.9

Rewrite

8

$8$ as

2^{2} \cdot 2

$2^{2} \cdot 2$ .

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Step 3.9.1

Factor

4

$4$ out of

8

$8$ .

D i s t a n c e = \sqrt{4 (2)}

$D i s t a n c e = \sqrt{4 (2)}$

Step 3.9.2

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

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

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

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

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

Step 3.10

Pull terms out from under the radical.

D i s t a n c e = 2 \sqrt{2}

$D i s t a n c e = 2 \sqrt{2}$

D i s t a n c e = 2 \sqrt{2}

$D i s t a n c e = 2 \sqrt{2}$

Step 4

The distance between

(1, 2, 3)

$(1, 2, 3)$ and

(3, 2, 1)

$(3, 2, 1)$ is

2 \sqrt{2}

$2 \sqrt{2}$ .

2 \sqrt{2} \approx 2.82842712

$2 \sqrt{2} \approx 2.82842712$

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