Question

In Exercises 75-78, describe the family of } & \text { planes }\end{array}\( represented by the equation, where \)c$ is any real number. $$ x+y+z=c $$

Short answer

The planes represented by the equation \(x + y + z = c\) constitute a family of parallel planes with same orientation, where \(\(c\)\) determines the distance of each plane from the origin.

Step-by-step solution

Understand the Plane Equation

The equation of a plane in three dimensions is given by \(Ax+By+Cz=D\), where \(A, B, C\) are the coefficients of \(x, y, z\) respectively and they are the normal to the plane, and \(D\) is the distance from the origin. In the given equation \(x+y+z=c\), the coefficients of \(x, y, z\) are all 1. So, the normal to the plane is \(<1,1,1>\) and \(c\) is the distance from the origin.

Effects of varying c

Altering the value of \(c\) changes the position of the plane in space, but not its orientation. When \(c\) varies, the resulting planes are all parallel to each other and have the same orientation because the normal to the plane (which determines its orientation) is not changing.

Describing the Family of Planes

So, the family of planes represented by the equation \(x+y+z=c\) consists of all planes that are parallel to each other (have the same orientation), with the vector \(<1,1,1>\) normal to the plane, and vary only in their distance from the origin.