SL 3.1—3d space, volume, angles, midpoints

In 2D, every point is located using two coordinates $(x,y)$. In three-dimensional space, we add a third axis , the $z$-axis , perpendicular to both the $x$- and $y$-axes. Every point in 3D space is then described by an ordered triple $(x,y,z)$.

Ordered Triple: A set of three numbers $(x,y,z)$ that uniquely identifies the position of a point in 3D space. The $x$- and $y$-values describe horizontal position (as in 2D), while the $z$-value describes height or depth.

Think of the corner of a room: the floor gives you $x$ and $y$, and the height up the wall gives you $z$. Any point in that room can be reached by moving along all three axes.

The distance formula in 3D is a direct extension of the 2D Pythagorean theorem , we simply include the difference in the $z$-coordinates as well.

3D Distance Formula: The distance $d$ between two points $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ in 3D space is:
$d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2+(z_2-z_1{)}^2}$

The logic: moving from point 1 to point 2, you travel $\Delta x$ in the $x$-direction, $\Delta y$ in the $y$-direction, and $\Delta z$ in the $z$-direction. The straight-line distance is the diagonal of the 3D rectangular box formed by these three steps , found using the Pythagorean theorem applied twice.