A point in three-dimensional space can be represented in a...

A point in three-dimensional space can be represented in a three-dimensional coordinate system. In such a case, a $z$ -axis is taken perpendicular to both the $x$ - and $y$ -axes. A point $A$ is assigned an ordered triple $A(x, y, z)$ relative to a fixed origin where the three axes meet. For Exercises $91-94$, determine the distance between the two given points in space. Use the distance formula $d=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}+\left(z_{2}-z_{1}\right)^{2}}$. (6,-4,-1) and (2,3,1)

A point in three-dimensional space can be represented in a three-dimensional coordinate system. In such a case, a $z$ -axis is taken perpendicular to both the $x$ - and $y$ -axes. A point $A$ is assigned an ordered triple $A(x, y, z)$ relative to a fixed origin where the three axes meet. For Exercises $91-94$, determine the distance between the two given points in space. Use the distance formula
$d=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}+\left(z_{2}-z_{1}\right)^{2}}$.
(6,-4,-1) and (2,3,1)

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Key Concepts

Distance Formula in Three Dimensions

The distance formula in three dimensions is an extension of the Pythagorean theorem and is used to compute the straight-line distance between two points in space. It is given by the square root of the sum of the squares of the differences between corresponding coordinates of the two points, effectively measuring the Euclidean distance in 3D.

Three-Dimensional Coordinate System

A three-dimensional coordinate system provides a framework to represent points in space using three axes, typically labeled x, y, and z. This system allows for the precise location of any point in space by assigning an ordered triple (x, y, z), which corresponds to its distances along each of the perpendicular axes intersecting at a common origin.

Ordered Triple

An ordered triple is a set of three numbers that uniquely identifies a point's position in a three-dimensional space. Each number in the triple corresponds to the coordinate along the x, y, and z axes respectively, allowing for a complete description of the point's location relative to the origin.

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