How to Find a Line Parallel to Two Planes?

Finding a line parallel to two planes involves understanding both the geometric and algebraic properties of planes and lines in three-dimensional space.

Step-by-Step Process

1. Identify the Normal Vectors
   Each plane in 3D space can be represented by an equation of the form:
   $Ax + By + Cz = D$
   The coefficients $A$, $B$, and $C$ form the normal vector $textbf{n} = (A, B, C)$ of the plane. Let’s say we have two planes with normal vectors $textbf{n}_1 = (A_1, B_1, C_1)$ and $textbf{n}_2 = (A_2, B_2, C_2)$

2. Find a Direction Vector Parallel to Both Planes
   A line parallel to both planes must be perpendicular to both normal vectors. To find such a direction vector, we can use the cross product of the two normal vectors:
   $textbf{d} = textbf{n}_1 times textbf{n}_2$
   The cross product of two vectors $textbf{a} = (a_1, a_2, a_3)$ and $textbf{b} = (b_1, b_2, b_3)$ is given by:
   $textbf{a} times textbf{b} = (a_2b_3 – a_3b_2, a_3b_1 – a_1b_3, a_1b_2 – a_2b_1)$

3. Define the Line Equation
   Once we have the direction vector $textbf{d}$, we need a point through which the line passes. Any point will work, but often the origin or a point of intersection of the planes (if they intersect) is used. The parametric form of the line is:
   $textbf{r}(t) = textbf{r}_0 + ttextbf{d}$
   where $textbf{r}_0$ is a point on the line, $textbf{d}$ is the direction vector, and $t$ is a parameter.

Example

Consider two planes with equations:

1. $2x + 3y – z = 5$
2. $x – y + 4z = 2$

1. Normal Vectors
   The normal vectors are:
   $textbf{n}_1 = (2, 3, -1)$
   $textbf{n}_2 = (1, -1, 4)$

2. Cross Product
   The direction vector $textbf{d} = textbf{n}_1 times textbf{n}_2$ is:
   $textbf{d} = (3*4 – (-1)(-1), (-1)(1) – 2*4, 2*(-1) – 3*1)$
   $textbf{d} = (12 – 1, -1 – 8, -2 – 3)$
   $textbf{d} = (11, -9, -5)$

3. Line Equation
   Using the origin (0, 0, 0) as a point on the line, the parametric equation is:
   $textbf{r}(t) = (0, 0, 0) + t(11, -9, -5)$
   $textbf{r}(t) = (11t, -9t, -5t)$

Conclusion

By following these steps, you can find a line parallel to two planes. This line will have a direction vector that is the cross product of the normal vectors of the planes, ensuring it is parallel to both planes.

Citations

1. 1. Khan Academy – Parallel and Perpendicular Planes
2. 2. MathWorld – Plane
3. 3. Paul’s Online Math Notes – Equations of Planes