Summary of Vectors in Three Dimensions

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

Key Equations

  • Distance between two points in space
    [latex]d=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}[/latex]
  • Sphere with center [latex](a,b,c)[/latex] and radius [latex]r[/latex]
    [latex](x-a)^{2}+(y-b)^{2}+(z-c)^{2}=r^{2}[/latex]

Glossary

coordinate plane
a plane containing two of the three coordinate axes in the three-dimensional coordinate system, named by the axes it contains: the [latex]xy[/latex]-plane, [latex]xz[/latex]-plane, or the [latex]yz[/latex]-plane
octants
the eight regions of space created by the coordinate planes
right-hand rule
a common way to define the orientation of the three-dimensional coordinate system; when the right hand is curved around the [latex]z[/latex]-axis in such a way that the fingers curl from the positive [latex]x[/latex]-axis to the positive [latex]y[/latex]-axis, the thumb points in the direction of the positive [latex]z[/latex]-axis
sphere
the set of all points equidistant from a given point known as the center
standard equation of a sphere
[latex](x-a)^{2}+(y-b)^{2}+(z-c)^{2}=r^{2}[/latex] describes a sphere with center [latex](a,b,c)[/latex] and radius [latex]r[/latex]
three-dimensional rectangular coordinate system
a coordinate system defined by three lines that intersect at right angles; every point in space is described by an ordered triple [latex](x,y,z)[/latex] that plots its location relative to the defining axes