7.3 Lines and Planes in 3D Space: Introduction to Notation

To define a line, we specify a point on it, [latex](x_0, y_0, z_0)[/latex], and a non-zero direction vector pointing along it [latex]\textbf{v} = (v_x, v_y, v_z)[/latex]. The equation of the line is then given by

\[(x, y, z) = (x_0, y_0, z_0) + s(v_x, v_y, v_z).\]

We read this as x y z is equal to, x zero y zero z zero plus, s times, v x v y v z.

A line can be represented in various forms. One such representation is the parametric form, given by

\[x = x_0 + s v_x, y = y_0 + s v_y, z = z_0 + s v_z.\]

We read this as x is equal to x zero plus s v x, y is equal to y zero plus s v y, z is equal to z zero plus s v z.

This form is called parametric because each coordinate is expressed separately as a function of the parameter [latex]s[/latex]. Notice that the three equations together are equivalent to the single vector equation given above, simply written out component by component.

Alternatively, if we solve each equation of the parametric form for $[latex]s[/latex]$, and set the resulting expressions equal, we obtain

\[\frac {x - x_0} {v_x} = \frac {y - y_0} {v_y} = \frac {z - z_0} {v_z},\]

which is known as the symmetric form of the line.

We read this as x minus x zero over v x is equal to, y minus y zero over v y which is equal to, z minus z zero over v z.

Just as a line is a one-dimensional object with no width, a plane is a two-dimensional object with no thickness, and both extend infinitely in their respective directions. Given a reference point [latex](x_0, y_0, z_0)[/latex] in a plane, and a normal vector perpendicular to the plane [latex]\textbf{n} = (a, b, c)[/latex], the equation of the plane can be written as

\[a(x - x_0) + b(y - y_0) + c(z - z_0) = 0.\]

We read these as a times, x minus x zero, plus b times, y minus y zero, plus c times, z minus z zero, is equal to zero.

This equation expresses the condition that, for any point [latex](x,y,z)[/latex] in the plane, the displacement vector [latex](x-x_0,y-y_0,z-z_0)[/latex] lies in the plane and is perpendicular to the normal vector [latex]\textbf{n} = (a, b, c)[/latex].

After rearranging, the equation of the plane takes the simpler standard form

\[a x + b y + c z = d.\]

We read this as a x plus b y plus c z is equal to d.

This is the form most commonly encountered in practice. Note that [latex]d = a x_0 + b y_0 + c z_0[/latex]. Furthermore, notice that the coefficients $[latex]a[/latex]$, $[latex]b[/latex]$, $[latex]c[/latex]$ are exactly the components of the normal vector, so reading off the normal vector from the equation of a plane is straightforward.

Given two different vectors u and non-zero v, we denote the projection of u along v by

\[ \text{proj}_\textbf{v} (\textbf{u}).\]

This is read out as the projection of u along v or the projection of u in the direction of v.