Question

Determine if any of the planes are parallel or identical. $$ \begin{array}{l} P_{1}:-60 x+90 y+30 z=27 \\ P_{2}: 6 x-9 y-3 z=2 \\ P_{3}:-20 x+30 y+10 z=9 \\ P_{4}: 12 x-18 y+6 z=5 \end{array} $$

Short answer

Planes \(P_{1}\) and \(P_{3}\) are identical. Planes \(P_{2}\) and \(P_{4}\) are parallel.

Step-by-step solution

Normalize the Planes

To compare the planes, first normalize them by dividing the entire equation of each plane by the coefficient of the z variable, thus obtaining the normalized equations: \(P_{1}\): -2x + 3y + z = 9/10, \(P_{2}\): 2x - 3y - z = -2/3, \(P_{3}\): -2x + 3y + z = 9/10, and \(P_{4}\): 2x - 3y + z = 5/6.

Compare the Normal Vectors

Recall that the normal vector for a plane with equation ax + by + cz = d is (a, b, c). For each of the normalized planes, the normal vector is \((a, b, 1)\). Comparing these vectors, we can see that \(P_{1}\), \(P_{3}\) have the same normal vectors, as well as \(P_{2}\) and \(P_{4}\). This means that \(P_{1}\) and \(P_{3}\) are parallel, as well as \(P_{2}\) and \(P_{4}\).

Check for Identical Planes

For two planes to be identical, they must be not only parallel, but the constant term on the right side of their equation must also have the same ratio as their normal vectors. Here \(P_{1}\) and \(P_{3}\) have the same right side and therefore are identical. While \(P_{2}\) and \(P_{4}\) have different constant terms and therefore are just parallel planes.

Key concepts

Normalizing Plane Equations

When you're dealing with multiple plane equations in calculus, normalizing these equations is a crucial step towards simplifying comparisons. Normalizing a plane equation involves dividing the entire equation by a specific coefficient to achieve a standard form. Most often, we choose to divide by the coefficient of the z-variable so that the z-coefficient becomes 1.

For example, if we start with the equation \( -60x + 90y + 30z = 27 \), normalizing by the z-coefficient (30 in this case) results in the simpler equation \( -2x + 3y + z = 9/10 \). This process is repeated for all plane equations in question.

By normalizing, we reduce complications when it comes to identifying relationships between planes. After normalizing, it's easier to compare and determine if the planes are parallel, identical, or neither. Remember, when normalizing, always maintain the balance of the equation—what you do to one side, do also to the other.

Comparing Normal Vectors

To understand the orientation of planes in space, we compare their normal vectors. A normal vector is perpendicular to the plane's surface and is unique to each plane. When the normal vectors of two planes are scalar multiples of each other, the planes are considered parallel.

If we take the simplified equations post-normalizing and identify the coefficients of x, y, and z, these coefficients form the normal vector of the plane. For instance, \( P_{1} \) normalized results in the coefficients -2, 3, and 1 which translates to the normal vector \( (-2, 3, 1) \).

Carrying out this process for all given planes, we can quickly compare the vectors to determine which planes are parallel. Parallelism in planes, much like in lines, indicates that the planes will never intersect no matter how far they extend into space. It's an intuitive concept that becomes much clearer with the visual aid of the vectors.

Identical Planes

When we have parallel planes with the same normal vectors, the next step is to determine if they are, in fact, identical—that is, the same plane represented with different equations. For two planes to be identical, they need to be parallel, and their equation's constant terms must also maintain the same ratio as the coefficients of the normal vector.

Considering our example, \( P_{1} \) and \( P_{3} \) not only share normal vectors but also have the same constant term when normalized \( (9/10) \). This matching ratio indicates that \( P_{1} \) and \( P_{3} \) are not merely parallel but indeed the same planar surface in space. In contrast, while \( P_{2} \) and \( P_{4} \) have the same directionality due to their normal vectors, differing constant terms mean they are parallel but distinct planes.

This concept is a fundamental part of understanding geometries in a three-dimensional space and arises frequently in fields such as computer graphics, physics, and engineering design.