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 normals are scalar multiples, perpendicular if the dot product of normals is zero.

Step 1: Identify the Normal Vectors

To determine if two planes are parallel or perpendicular, we first need to identify their normal vectors. The normal vector for the first plane \(A_1x + B_1y + C_1z = D_1\) is \(\mathbf{n_1} = \langle A_1, B_1, C_1 \rangle\) and for the second plane \(A_2x + B_2y + C_2z = D_2\) is \(\mathbf{n_2} = \langle A_2, B_2, C_2 \rangle\).

Step 2: Check for Parallelism

Two planes are parallel if their normal vectors are scalar multiples of each other. That means \(\mathbf{n_1} = k \cdot \mathbf{n_2}\) for some scalar \(k\). Algebraically, this gives the equations: \(A_1 = kA_2\), \(B_1 = kB_2\), \(C_1 = kC_2\). If such a \(k\) exists, the planes are parallel.

Step 3: Verify Perpendicularity

Two planes are perpendicular if their normal vectors are orthogonal. This occurs when the dot product of the two normal vectors is zero: \(\mathbf{n_1} \cdot \mathbf{n_2} = A_1A_2 + B_1B_2 + C_1C_2 = 0\). If this equation is satisfied, the planes are perpendicular.

Key Concepts

Normal Vectors

In the context of planes in space, a crucial concept to understand is the normal vector. Each plane can be represented by an equation of the form \(A x + B y + C z = D\). Here, the coefficients \(A\), \(B\), and \(C\) represent the components of the normal vector for that plane. The normal vector is denoted as \(\mathbf{n} = \langle A, B, C \rangle\). It is a vector that is perpendicular to the surface of the plane, much like a spear sticking out from a shield. This directional property means the normal vector dictates the orientation of the plane in space.

When dealing with multiple planes, identifying their normal vectors is the first step in determining relationships such as parallelism and perpendicularity. Essentially, the normal vector acts as a fingerprint for the plane's orientation, providing insight into how it aligns or intersects with other planes.

Parallel Planes

To determine if two planes are parallel, we examine their normal vectors. Parallel planes have normal vectors that are scalar multiples of one another. This means if you take the normal vector of one plane and multiply it by a common constant (scalar), you should be able to align it with the normal vector of the other plane.

Mathematically, if two planes have equations \(A_1 x + B_1 y + C_1 z = D_1\) and \(A_2 x + B_2 y + C_2 z = D_2\), they are parallel if there's a scalar \(k\) such that:

• \(A_1 = k A_2\)
• \(B_1 = k B_2\)
• \(C_1 = k C_2\)

If you can find such a \(k\), then the two planes never intersect, meaning they extend infinitely without ever meeting, essentially mirroring each other's direction in space.

Perpendicular Planes

Planes are perpendicular when their normal vectors are orthogonal. This concept involves the dot product of the vectors. Two vectors are orthogonal if their dot product is zero.

For two planes given by the equations \(A_1 x + B_1 y + C_1 z = D_1\) and \(A_2 x + B_2 y + C_2 z = D_2\), we check their perpendicularity by calculating the dot product of their normal vectors \(\mathbf{n_1} = \langle A_1, B_1, C_1 \rangle\) and \(\mathbf{n_2} = \langle A_2, B_2, C_2 \rangle\). The calculation is as follows:

\(A_1 A_2 + B_1 B_2 + C_1 C_2 = 0\)

If this condition is met, the planes intersect at right angles. This makes perpendicular planes crucial in various designs and structures, ensuring stiff and stable framework by relying on right angle interactions.