Problem 72 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the equation of the plane passing through the origin and parallel to (a) the \(x y\)-plane (b) the plane \(x+y+z=1\)

Step-by-Step Solution

Answer

a) The equation is z = 0. b) The equation is x + y + z = 0.

Step 1: Understanding the Plane through the Origin

Any plane that passes through the origin can be represented by the equation \(ax + by + cz = 0\). This ensures that the origin \( (0,0,0) \) is a solution to the equation. Our goal is to find the specific coefficients \(a, b, c\) for each part of the exercise.

Step 2: Equation of Plane parallel to the xy-plane

A plane parallel to the \(xy\)-plane has a normal vector parallel to the \(z\)-axis; thus, only the \('c'\) component of the normal vector is non-zero. Therefore, the equation of a plane parallel to the \(xy\)-plane and passing through the origin is \(z = 0\) or equivalently \(0x + 0y + 1z = 0\).

Step 3: Deriving the Normal Vector of a Parallel Plane

A plane parallel to another plane will have a normal vector that is a scalar multiple of the normal vector of the given plane. The normal vector of the plane \(x+y+z=1\) is \(\langle 1, 1, 1 \rangle\). So, a parallel plane through the origin will have the same normal vector \(\langle 1, 1, 1 \rangle\).

Step 4: Equation of Plane parallel to x+y+z=1

Using the normal vector \(\langle 1, 1, 1 \rangle\), the equation of the plane passing through the origin and parallel to \(x+y+z=1\) is \(x+y+z=0\). This ensures that the plane maintains the orientation of the given plane but is shifted to pass through the origin.

Key Concepts

Normal Vector

In geometry, a normal vector is an essential concept when dealing with planes, as it defines the orientation of the plane in three-dimensional space. A normal vector is a vector that is perpendicular, or "normal," to every line in the plane it represents. This vector is crucial for constructing plane equations because it provides a way to describe the unique direction of a plane.

For example, consider a plane given by the equation \(ax + by + cz = d\). The coefficients \(a, b,\) and \(c\) specify a normal vector \(\langle a, b, c \rangle\) for the plane. This vector sets the direction in which the plane "faces."

When working with parallel planes, understanding the role of the normal vector becomes even more significant. Parallel planes share a normal vector, or one that is a scalar multiple, highlighting their aligned orientations and equal tilts in space.

Parallel Planes

Parallel planes are akin to parallel lines: they never meet, no matter how far along the infinite space you extend them. This unique property of parallel planes is due to their identical normal vectors. When two planes have the same or scalar-multiples of each other's normal vectors, they are parallel to each other.

An example of parallel planes would be the \(xy\)-plane and any plane parallel to it, both having normal vectors that indicate no deviation along the \(z\)-axis, like \(\langle 0, 0, 1 \rangle\). Meanwhile, if you have a plane like \(x+y+z=1\), any parallel plane would have a normal vector such as \(\langle 1, 1, 1 \rangle\), maintaining the exact orientation.

To determine the equation of a plane parallel to a given plane, keep the normal vector constant while changing the constant on the right side of the equation. For instance, if \(x + y + z = 1\) is one plane, a parallel plane passing through the origin is \(x + y + z = 0\), ensuring the same angle and tilt in space.

Origin in Coordinate Geometry

Coordinate geometry often involves identifying and working with planes that intersect through specific points, with the origin \((0,0,0)\) being the most central point in a Cartesian coordinate system. Planes that pass through the origin have equations where the constant term is zero, making the origin a trivial solution.

A plane's equation typically takes the form \(ax + by + cz = 0\) when it must go through the origin. This removal of the constant term ensures that the origin remains on the plane while dictifying the combination of other variables that satisfy the plane's orientation.

Understanding planes through the origin also provides insight into how plane orientation can change relative to the axes, which assists in solving exercises involving rotations and translations of geometric figures within the coordinate system.