Vectors in Three-Dimensional Space

In three-dimensional space, we add a $z$-axis perpendicular (orthogonal) to the usual $x$- and $y$-axis to create a rectangular coordinate system called $xyz$-space. The $z$-axis is oriented relative to the $xy$-plane in accordance to the right-hand rule. Any pair of axes span a coordinate plane, and these coordinate planes split $xyz$-space into eight octants. Each point can be described by a ordered triple $(x,y,z).$ All computations from two-dimensional space regarding either points or vectors generalize to three-dimensional space in the most natural way.

Given two points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2),$ the distance between them is $\sqrt{(x_2\!-\!x_1)^2+(y_2\!-\!y_1)^2+(z_2\!-\!z_1)^2},$ and their midpoint has coordinates $(\overline{x}, \overline{y}, \overline{z})$ given by the formulas $\overline{x} = \frac{1}{2}(x_1\!+\!x_2)$ and $\overline{y} = \frac{1}{2}(y_1\!+\!y_2)$ and $\overline{z} = \frac{1}{2}(z_1\!+\!z_2).$ In three-dimensional space, two points uniquely determine a line and three non-colinear points uniquely determine a plane. A sphere of radius $r$ centered at $(h,k,\ell)$ has equation $r^2 = (x_2-h)^2+(y_2-k)^2+(z_2-\ell)^2\,.$ If a sphere intersects a plane in space, the resulting shape of their intersection, generically a circle, is called the trace of the sphere in the plane.

Vectors in $xyz$-space have three components. The magnitude of $\bm{v} = \langle v_1, v_2, v_3\rangle$ is calculated as $|\bm{v}| = \sqrt{v_1^2+v_2^2+v_3^2}\,.$ In addition to the unit coordinate vectors ${\mathbf{i} = \langle 1,0,0 \rangle}$ and ${\mathbf{j} = \langle 0,1,0 \rangle,}$ we now have ${\mathbf{k} = \langle 0,0,1 \rangle}.$ For vectors ${\bm{u} = \langle u_1,u_2,u_3 \rangle}$ and ${\bm{v} = \langle v_1,v_2,v_3 \rangle}$ their dot product is calculated as ${\bm{u}\cdot\bm{v} = u_1v_1 + u_2v_2 + u_3v_3} = |\bm{u}||\bm{v}|\cos(\theta).$ The direction angles of a vector $\bm{v}$ are the three angles between $\bm{v}$ and each of the three coordinate axes. The cosines of these angles are referred to as the vector’s direction cosines. If $\alpha$ and $\beta$ and $\gamma$ are the direction angles of $\bm{v}$ with the $x$- and $y$- and $z$-axis respectively, then $$\cos(\alpha) = \frac{v_1}{|\bm{v}|} \quad \cos(\beta) = \frac{v_2}{|\bm{v}|} \quad \cos(\gamma) = \frac{v_2}{|\bm{v}|} \qquad \bm{v} = \Bigl\langle |\bm{v}|\cos(\alpha), |\bm{v}|\cos(\beta), |\bm{v}|\cos(\gamma) \Bigr\rangle \,.$$