In three-dimensional space, we add a \(z\)-axis orthogonal to the \(x\)- and \(y\)-axis. Any pair of axes span a coordinate plane, and these coordinate planes split space into eight octants. [see first section] Each point can be described by a ordered triple \((x,y,z).\)
Simple planes like y=5.
The distance between two points \(P(x_1, y_1, z_1)\) and \(Q(x_2, y_2, z_2)\) in three-dimensional space is \[ \mathrm{d}(P,Q) = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\,. \] [discuss midpoint formula too?] Then 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, generically a circle, is the trace of the sphere in the plane.
Vectors in three-dimensional space are about the same as in two dimensional space; now they have three components. The magintude of \(\bm{v} = \langle a,b,c\rangle\) is now \[|\bm{v} = \sqrt{a^2+b^2+c^2}\,.\]
sums, differences, and scalar multiples of vectors. Component vectors \(\mathbm{i}\) and \(\mathbm{j}\) and \(\mathbm{k}.\)
Dot product is the same. Angle between vectors is the same.
oof direction angles. and direction cosines.