SL 3.1—Three-dimensional geometry

In three-dimensional coordinates, a point is (x,y,z). A displacement vector records the change in each coordinate, so distance and direction can be calculated component by component.

For points P and Q, PQ=Q−P and |PQ|=√((Δx)²+(Δy)²+(Δz)²). The midpoint averages corresponding coordinates. These formulas are geometric statements, not just calculator rules.

From P(1,−2,3) to Q(4,2,−1), the displacement is (3,4,−4) and the distance is √41. The sign tells direction; squaring removes it only when measuring length.

Do not drop the z-coordinate or confuse a displacement component with the total distance. Keep units and point order clear when interpreting direction.

The midpoint of $P(x_1,y_1,z_1)$ and $Q(x_2,y_2,z_2)$ is $((x_1+x_2)/2,(y_1+y_2)/2,(z_1+z_2)/2)$. For solids, use the correct base area and perpendicular height: prism $V=Ah$, pyramid or cone $V=Ah/3$, sphere $V=4\pi r^3/3$; total surface area includes every exposed face. In a cuboid, identify a face diagonal first, then use a second right triangle to obtain a space length or an angle between a line and a plane.