Question

Describe the family of } & \text { planes }\end{array}\( represented by the equation, where \)c$ is any real number. $$ c y+z=0 $$

Short answer

The equation \(cy + z = 0\) represents a family of planes in 3D space, all of which are parallel to the x-axis and their orientation changes with the value of \(c\).

Step-by-step solution

Understand the Equation of a Plane

The equation of a plane in three dimensions is usually given in the form \(ax + by + cz = d\), where \([a, b, c]\) are the direction cosines of the normal vector to the plane. If any two of \([a, b, c]\) are zero, the plane is parallel to the corresponding axis.

Relate the given equation to the standard form

The given equation is \(cy + z = 0\). We can rewrite it in the standard form by dividing every term by \(c\) (unless \(c = 0\)), obtaining \(y + (1/c)z = 0\). We can recognize that 'a' and 'c' in the standard equation matches 0 and 1/c (from the rewritten equation, respectively), while 'b' matches 1. Hence, the plane is parallel to the x-axis and the orientation of the plane in space changes with c.

Interpret the Influence of c

The coefficient \(c\) basically describes the 'slope' of the plane, that is how much it inclines with respect to the yz-plane. If \(c = 0\), we have the original yz-plane. As \(c\) moves away from zero, the plane rotates about the x-axis. The positive or negative value of \(c\) determines the direction of this rotation.