Essential Concepts

• The three-dimensional coordinate system is built around a set of three axes that intersect at right angles at a single point, the origin. Ordered triples $(x,y,z)$ are used to describe the location of a point in space.
• The distance $d$ between points $(x_{1},y_{1},z_{1})$ and $(x_{2},y_{2},z_{2})$ is given by the formula $d=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}$.
• In three dimensions, the equations $x=a$, $y=b$, and $z=c$ describe planes that are parallel to the coordinate planes.
• The standard equation of a sphere with center $(a,b,c)$ and radius $r$ is $(x-a)^{2}+(y-b)^{2}+(z-c)^{2}=r^{2}$.
• In three dimensions, as in two, vectors are commonly expressed in component form, ${\bf{v}}=\langle{x,y,z}\rangle$, or in terms of the standard unit vectors, $x{\bf{i}}+y{\bf{j}}+z{\bf{k}}$.
• Properties of vectors in space are a natural extension of the properties for vectors in a plane. Let ${\bf{v}}=\langle{x_{1},y_{1},z_{1}}\rangle$ and ${\bf{w}}=\langle{x_{2},y_{2},z_{2}}\rangle$ be vectors, and let $k$ be a scalar.
  • Scalar multiplication: $k{\bf{v}}=\langle{kx_{1},ky_{1},kz_{1}}\rangle$
  • Vector addition: ${\bf{v}}+{\bf{w}}=\langle{x_{1},y_{1},z_{1}}\rangle+\langle{x_{2},y_{2},z_{2}}\rangle=\langle{x_{1}+x_{2},y_{1}+y_{2},z_{1}+z_{2}}\rangle$
  • Vector subtraction: ${\bf{v}}-{\bf{w}}=\langle{x_{1},y_{1},z_{1}}\rangle-\langle{x_{2},y_{2},z_{2}}\rangle=\langle{x_{1}-x_{2},y_{1}-y_{2},z_{1}-z_{2}}\rangle$
  • Vector magnitude: $\parallel{\bf{v}}\parallel=\sqrt{{x_{1}}^{2}+{y_{1}}^{2}+{z_{1}}^{2}}$
  • Unit vector in the direction of ${\bf{v}}$: $\frac{\bf{v}}{\parallel{\bf{v}}\parallel}=\frac{1}{\parallel{\bf{v}}\parallel}\langle{x_{1},y_{1},z_{1}}\rangle=\langle{\frac{x_{1}}{\parallel{\bf{v}}\parallel},\frac{y_{1}}{\parallel{\bf{v}}\parallel},\frac{z_{1}}{\parallel{\bf{v}}\parallel}}\rangle,{\bf{v}}\ne{0}$