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Intro

This formula gives the distance between two points in 3d space.

Notes

Distance Formula 3D

\( \text{Distance} = \sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2} \)

Practice Problems

\(\textbf{1)}\) Find the distance between \((-1,0,1) \text{ and } (2,0,5)\)

Show Distance

The distance between the points is \(5\)

Show Work

\(\,\,\,\,\,\text{Distance} = \sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2}\)

\(\,\,\,\,\,\text{Distance} = \sqrt{((-1)-(2))^2+((0)-(0))^2+((1)-(5))^2}\)

\(\,\,\,\,\,\text{Distance} = \sqrt{(-3)^2+(0)^2+(-4)^2}\)

\(\,\,\,\,\,\text{Distance} = \sqrt{9+0+16}\)

\(\,\,\,\,\,\text{Distance} = \sqrt{25}\)

\(\,\,\,\,\,\text{Distance} = 5\)

\(\textbf{2)}\) Find the distance between \((-4,0,-12) \text{ and } (2,8,12)\)

Show Distance

The distance between the points is \(26\)

Show Work

\(\,\,\,\,\,\text{Distance} = \sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2}\)

\(\,\,\,\,\,\text{Distance} = \sqrt{((-4)-(2))^2+((0)-(8))^2+((-12)-(12))^2}\)

\(\,\,\,\,\,\text{Distance} = \sqrt{(-6)^2+(-8)^2+(-24)^2}\)

\(\,\,\,\,\,\text{Distance} = \sqrt{36+64+576}\)

\(\,\,\,\,\,\text{Distance} = \sqrt{676}\)

\(\,\,\,\,\,\text{Distance} = 26\)

\(\textbf{3)}\) Find the distance between \((1,2,3) \text{ and } (4,5,6)\)

Show Distance

The distance between the points is \(3\sqrt{3}\)

Show Work

\(\,\,\,\,\,\text{Distance} = \sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2}\)

\(\,\,\,\,\,\text{Distance} = \sqrt{((1)-(4))^2+((2)-(5))^2+((3)-(6))^2}\)

\(\,\,\,\,\,\text{Distance} = \sqrt{(-3)^2+(-3)^2+(-3)^2}\)

\(\,\,\,\,\,\text{Distance} = \sqrt{9+9+9}\)

\(\,\,\,\,\,\text{Distance} = \sqrt{27}\)

\(\,\,\,\,\,\text{Distance} = 3\sqrt{3}\)

\(\textbf{4)}\) Find the distance between \((4,2,3) \text{ and } (5,5,5)\)

Show Distance

The distance between the points is \(\sqrt{14}\)

Show Work

\(\,\,\,\,\,\text{Distance} = \sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2}\)

\(\,\,\,\,\,\text{Distance} = \sqrt{((4)-(5))^2+((2)-(5))^2+((3)-(5))^2}\)

\(\,\,\,\,\,\text{Distance} = \sqrt{(-1)^2+(-3)^2+(-2)^2}\)

\(\,\,\,\,\,\text{Distance} = \sqrt{1+9+4}\)

\(\,\,\,\,\,\text{Distance} = \sqrt{14}\)

\(\textbf{5)}\) Find the distance between \((1,3,-4) \text{ and } (2,2,6)\)

Show Distance

The distance between the points is \(\sqrt{102}\)

Show Work

\(\,\,\,\,\,\text{Distance} = \sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2}\)

\(\,\,\,\,\,\text{Distance} = \sqrt{((1)-(2))^2+((3)-(2))^2+((-4)-(6))^2}\)

\(\,\,\,\,\,\text{Distance} = \sqrt{(-1)^2+(1)^2+(-10)^2}\)

\(\,\,\,\,\,\text{Distance} = \sqrt{1+1+100}\)

\(\,\,\,\,\,\text{Distance} = \sqrt{102}\)

\(\,\,\,\,\,\text{Distance} = \sqrt{102}\)

See Related Pages\(\)

\(\bullet\text{ Geometry Homepage}\)
\(\,\,\,\,\,\,\,\,\text{All the Best Topics…}\)

\(\bullet\text{ Rectangular Prisms- Volume}\)
\(\,\,\,\,\,\,\,\,\)\(V=l \cdot w \cdot h…\)

\(\bullet\text{ Rectangular Prisms- Surface Area}\)
\(\,\,\,\,\,\,\,\,\)\(SA=lw+wh+lh…\)

\(\bullet\text{ Diagonal of a Prism}\)
\(\,\,\,\,\,\,\,\,\)\(d=\sqrt{l^2+w^2+h^2}…\)

\(\bullet\text{ Cylinders- Volume and Surface Area}\)
\(\,\,\,\,\,\,\,\,\)\(V=\pi r^2h\,\,\,SA=2\pi r^2+2 \pi rh…\)

\(\bullet\text{ Pyramids- Volume and Surface Area}\)
\(\,\,\,\,\,\,\,\,\)\(V=\frac{1}{3}Bh\,\,\,SA=B+\frac{pl}{2}…\)

\(\bullet\text{ Cones- Volume and Surface Area}\)
\(\,\,\,\,\,\,\,\,\)\(V=\frac{1}{3}\pi r^2 h\,\,\,SA=\pi r^2+\pi r l…\)

\(\bullet\text{ Spheres- Volume and Surface Area}\)
\(\,\,\,\,\,\,\,\,\)\(V=\frac{4}{3}\pi r^3 \,\,\,SA=4 \pi r^2…\)

\(\bullet\text{ Similar figures}\)
\(\,\,\,\,\,\,\,\,\text{Similarity ratio } a:b, \text{Area ratio } a^2:b^2, \text{Volume ratio } a^3:b^3\)

\(\bullet\text{ Nets of Polyhedra}\)
\(\,\,\,\,\,\,\,\,\)

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