Essential Concepts

• In three dimensions, the direction of a line is described by a direction vector. The vector equation of a line with direction vector [latex]{\bf{v}}=\langle{a,b,c}\rangle[/latex] passing through point [latex]P=(x_{0},y_{0},z_{0})[/latex] is [latex]{\bf{r}}={\bf{r}}_{0}+t{\bf{v}}[/latex], where [latex]{\bf{r}}_{0}=\langle{x_{0},y_{0},z_{0}}\rangle[/latex] is the position vector of point [latex]P[/latex] This equation can be rewritten to form the parametric equations of the line: [latex]x=x_{0}+ta[/latex], [latex]y=y_{0}+tb[/latex], and [latex]z=z_{0}+tc[/latex].  The line can also be described with the symmetric equations [latex]\frac{x-x_{0}}{a}=\frac{y-y_{0}}{b}=\frac{z-z_{0}}{c}[/latex].
• Let [latex]L[/latex] be a line in space passing through point [latex]P[/latex] with direction vector [latex]{\bf{v}}[/latex]. If [latex]Q[/latex] is any point not on [latex]L[/latex] then the distance from [latex]Q[/latex] to [latex]L[/latex] is [latex]d=\frac{\parallel\overrightarrow{PQ}\times{\bf{v}}\parallel}{\parallel{\bf{v}}\parallel}[/latex].
• In three dimensions, two lines may be parallel but not equal, equal, intersecting, or skew.
• Given a point [latex]P[/latex] and vector [latex]{\bf{n}}[/latex] the set of all points [latex]Q[/latex] satisfying equation [latex]{\bf{n}}\cdot\overrightarrow{PQ}=0[/latex] forms a plane. Equation [latex]{\bf{n}}\cdot\overrightarrow{PQ}=0[/latex] is known as the vector equation of a plane.
• The scalar equation of a plane containing point [latex]P=(x_{0},y_{0},z_{0})[/latex] with normal vector [latex]{\bf{n}}=\langle{a,b,c}\rangle[/latex] is [latex]a(x-x_{0})+b(y-y_{0})+c(z-z_{0})=0[/latex]. This equation can be expressed as [latex]ax+by+cz+d=0[/latex], where [latex]d=-ax_{0}-by_{0}-cz_{0}[/latex]. This form of the equation is sometimes called the general form of the equation of a plane.
• Suppose a plane with normal vector [latex]{\bf{n}}[/latex] passes through point [latex]Q[/latex]. The distance [latex]D[/latex] from the plane to point [latex]P[/latex] not in the plane is given by [latex]D=\parallel\text{proj}_{\bf{n}}\overrightarrow{QP}\parallel=|\text{comp}_{\bf{n}}\overrightarrow{QP}|=\dfrac{|\overrightarrow{QP}\cdot{\bf{n}}|}{\parallel{\bf{n}}\parallel}[/latex].
• The normal vectors of parallel planes are parallel. When two planes intersect, they form a line.
• The measure of the angle [latex]\theta[/latex] between two intersecting planes can be found using the equation: [latex]\cos\theta=\frac{|{\bf{n}}_{1}\cdot{\bf{n}}_{2}|}{\parallel{\bf{n}}_{1}\parallel\parallel{\bf{n}}_{2}\parallel}[/latex], where [latex]{\bf{n}}_{1}[/latex] and [latex]{\bf{n}}_{2}[/latex] are normal vectors to the planes.
• The distance [latex]D[/latex] from the point [latex](x_{0},y_{0},z_{0})[/latex] to plane [latex]ax+by+cz+d=0[/latex] is given by [latex]D=\dfrac{|a(x_{0}-x_{1})+b(y_{0}-y_{1})+c(z_{0}-z_{1})|}{\sqrt{a^2+b^2+c^2}}=\dfrac{|ax_{0}+by_{0}+cz_{0}+d|}{\sqrt{a^2+b^2+c^2}}[/latex]