Problem 52 - Advanced Engineering Mathematics

Question

In Problems, find an equation of the plane that satisfies the given conditions. Contains the origin and is parallel to \(5 x-y+z=6\)

Step-by-Step Solution

Answer

The equation of the plane is \(5x - y + z = 0\).

Step 1: Understand the Problem

We are tasked with finding an equation of a plane that contains the origin and is parallel to a given plane. The given plane's equation is \(5x - y + z = 6\).

Step 2: Identify Plane Properties

A plane parallel to another shares the same normal vector. For the equation \(5x - y + z = 6\), the normal vector is \((5, -1, 1)\). Our target plane will have this same normal vector.

Step 3: Use the Plane Equation Form

Using the standard equation form of a plane, \(Ax + By + Cz = D\), where \((A, B, C)\) is the normal vector. Here, \(A=5\), \(B=-1\), and \(C=1\). Our equation is \(5x - y + z = D\).

Step 4: Determine the Constant Term

Since the plane contains the origin \((0, 0, 0)\), we substitute these values into the equation \(5x - y + z = D\). This gives us \(5(0) - (0) + (0) = D\), so \(D = 0\).

Step 5: Write the Equation of the Plane

Based on our calculations, the equation of the plane is \(5x - y + z = 0\), which contains the origin and is parallel to the given plane.

Key Concepts

Normal Vector

The normal vector is a critical component in understanding and working with planes in three-dimensional space. Think of the normal vector as a directional arrow sticking out of the plane at a 90-degree angle. This vector is unique because it defines the orientation of the plane.
In the equation of a plane given by \(Ax + By + Cz = D\), the normal vector is represented by the coordinates \((A, B, C)\). These values indicate the vector's direction in the space.

• For example, in the plane equation \(5x - y + z = 6\), the normal vector is \((5, -1, 1)\).
• This vector tells us that the plane tilts in the direction of '5' in the x-axis, '-1' in the y-axis, and '1' in the z-axis.

Once you know the normal vector, you can perform many useful calculations, like determining if another plane is parallel or finding intersections.

Parallel Planes

Parallel planes are fascinating because, although they never intersect, they share a special relationship. Two planes are parallel when their normal vectors point in the same direction or are scalar multiples of each other.
If you understand this relationship, you can see why parallel planes must have identical or proportional normal vectors.

• Given the plane equation \(5x - y + z = 6\), the normal vector is \((5, -1, 1)\).
• A parallel plane will also have a normal vector \((5, -1, 1)\), regardless of the constant \(D\) value in its equation form \(Ax + By + Cz = D\).

Thus, by knowing the normal vector, you can easily write the equation of a plane parallel to another by adjusting only the constant \(D\), so that it meets any additional conditions, such as containing a specific point.

Plane Equation Formulation

Formulating a plane equation requires understanding the geometric properties that define its position and orientation in space. The standard form for a plane equation is \(Ax + By + Cz = D\), where \((A, B, C)\) forms the normal vector.

• To derive a plane equation, identify the normal vector from a given condition or parallel plane. For example, from \(5x - y + z = 6\), the normal vector is \((5, -1, 1)\).
• Once the normal vector is known, ensure the plane satisfies any specified conditions, like passing through a particular point.

To formulate the new plane equation, substitute into \(Ax + By + Cz = D\) the coordinates of any point the plane should pass through. For a condition like containing the origin, set \((x, y, z) = (0, 0, 0)\), simplifying to find \(D = 0\).
Thus, you construct a new plane that meets all criteria while maintaining the correct orientation through its normal vector.