Problem 68 - Thomas Calculus

Question

How can you tell when two planes \(A_{1} x+B_{1} y+C_{1} z=D_{1}\) and \(A_{2} x+B_{2} y+C_{2} z=D_{2}\) are parallel? Perpendicular? Give reasons for your answer.

Step-by-Step Solution

Answer

Planes are parallel if their normal vectors are proportional; perpendicular if their dot product is zero.

Step 1: Understanding Plane Equations

Planes are described by the equations \(A_{1} x + B_{1} y + C_{1} z = D_{1}\) and \(A_{2} x + B_{2} y + C_{2} z = D_{2}\). Each plane has a normal vector given by the coefficients of \(x, y, z\). For the first plane, the normal vector is \(\mathbf{n}_1 = \langle A_1, B_1, C_1 \rangle\) and for the second plane, \(\mathbf{n}_2 = \langle A_2, B_2, C_2 \rangle\). Understanding these normal vectors is key to determining if the planes are parallel or perpendicular.

Step 2: Determining Parallel Planes

Two planes are parallel if their normal vectors are proportional. This means there exists a constant \(k\) such that \(\mathbf{n}_1 = k \times \mathbf{n}_2\). Hence, \(A_1 = k A_2\), \(B_1 = k B_2\), and \(C_1 = k C_2\). If these conditions are satisfied, the planes are parallel.

Step 3: Determining Perpendicular Planes

Two planes are perpendicular if the dot product of their normal vectors is zero. The dot product \(\mathbf{n}_1 \cdot \mathbf{n}_2\) is given by \(A_1 A_2 + B_1 B_2 + C_1 C_2 = 0\). If this equation holds true, it implies the planes are perpendicular.

Key Concepts

Understanding Normal Vectors

In the context of plane geometry, normal vectors play a crucial role in defining the orientation of a plane in three-dimensional space. Given a plane equation of the form \(A x + B y + C z = D\), the normal vector is represented as \(\langle A, B, C \rangle\). This vector is perpendicular to the plane, just like a flagpole standing upright on the ground. It specifies the direction in which the plane is facing.

The normal vector offers important insights:

• It helps determine if two planes are parallel or perpendicular.
• It can be used to find the angle between two planes.
• Its magnitude is not usually the focus, since its direction is what's more informative in these relations.

By understanding the role of normal vectors, students can solve various problems related to plane orientation and interaction in space.

Identifying Parallel Planes

Two planes are parallel when their normal vectors are proportional to each other. If two planes are described by the normal vectors \(\mathbf{n}_1 = \langle A_1, B_1, C_1 \rangle\) and \(\mathbf{n}_2 = \langle A_2, B_2, C_2 \rangle\), they are parallel if there exists a constant \(k\) such that:
- \(A_1 = kA_2\)
- \(B_1 = kB_2\)
- \(C_1 = kC_2\)

This means the directions given by these vectors are identical, implying the planes never intersect. Think of this condition like train tracks running parallel across a flat landscape. Regardless of their position, they maintain the same direction, ensuring they never meet.

Recognizing this condition is essential for solving geometry problems involving parallel relationships, making normal vectors indispensable in calculations.

Recognizing Perpendicular Planes

Perpendicular planes have normal vectors that meet at right angles to each other. To establish if two planes are perpendicular, we use the dot product of their normal vectors. For normal vectors \(\mathbf{n}_1 = \langle A_1, B_1, C_1 \rangle\) and \(\mathbf{n}_2 = \langle A_2, B_2, C_2 \rangle\), they are perpendicular if:
\[ A_1 A_2 + B_1 B_2 + C_1 C_2 = 0 \]
This is akin to forming an 'L' shape, much like the intersection of walls where they meet the ceiling. The planes intersect at right angles, each defining a distinct directional boundary.

By easily checking this dot product condition, students gain powerful insight into geometric relationships, helping solve diverse problems in plane geometry.