Problem 53 - Advanced Engineering Mathematics

Question

In Problems, find an equation of the plane that satisfies the given conditions. Contains \((3,6,12)\) and is parallel to the \(x y\) -plane

Step-by-Step Solution

Answer

The equation of the plane is \(z = 12\).

Step 1: Understand the Problem

We need to find the equation of a plane that contains the point \((3, 6, 12)\) and is parallel to the \(xy\)-plane. Planes parallel to the \(xy\)-plane have a normal vector that can be described as \(\boldsymbol{n} = \langle 0,0,1 \rangle\).

Step 2: Use the Plane Equation Formula

The general equation of a plane is given by \(Ax + By + Cz = D\), where \(\langle A, B, C \rangle\) is the normal vector of the plane. Since the plane is parallel to the \(xy\)-plane, the normal vector will be \(\boldsymbol{n} = \langle 0,0,1 \rangle\). Thus, the plane equation simplifies to \(z = D\).

Step 3: Substitute the Given Point

Substitute the point \((3, 6, 12)\) into the simplified plane equation \(z = D\). The se value of \(z\) from the point is \(12\), so we set \(D = 12\), giving us the equation \(z = 12\).

Step 4: Write the Final Equation

Since the plane must contain the point \((3, 6, 12)\) and be parallel to the \(xy\)-plane, the equation of the plane becomes \(z = 12\). This is because the z-coordinate is the only variable due to the parallel orientation with respect to the \(xy\)-plane.

Key Concepts

Normal Vector

The normal vector is an essential component in understanding plane equations and their orientations. A vector is normal, or perpendicular, to a surface when it hits the plane at a right angle. For any plane, the normal vector provides crucial information, as it defines the direction directly opposite the surface or plane.

• The general equation of a plane is given by \(Ax + By + Cz = D\), where \( \langle A, B, C \rangle \) is the normal vector.
• The normal vector’s components \(A\), \(B\), and \(C\) typically represent the coefficients of the variables \(x\), \(y\), and \(z\) respectively in the plane equation.

Understanding the normal vector is crucial for identifying the plane's orientation in 3D space. In our exercise, since the plane is parallel to the \(xy\)-plane, the normal vector is \( \langle 0, 0, 1 \rangle \). This vector points directly up along the \(z\)-axis, indicating that the \(z\)-coordinate is the key defining variable in the plane equation.

Parallel Planes

When talking about parallel planes, we're referring to two planes that never intersect. They always have the same distance apart, just like parallel lines. This is because their normal vectors are either equal or scalar multiples of one another, meaning they point in the same direction.

• If two planes are parallel, they can't share a common line or point, except at infinity.
• The normal vector plays an important role here—two planes are parallel if and only if their normal vectors are parallel.

In the given exercise, the plane described is parallel to the \(xy\)-plane. This specific orientation means that the normal vector \( \langle 0, 0, 1 \rangle \) is aligned along the \(z\)-axis, maintaining a constant distance from any point in the \(z\)-direction. Thus, the plane equation simplifies to focusing on the height \(z = D\), indicating that all points share the same \(z\)-coordinate, \(z=12\).

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. In three-dimensional space, points are defined using three coordinates \(x, y, \) and \(z\), which makes it possible to describe points and planes mathematically.

• In 3D geometry, the location of any plane or point can be precisely described with these coordinates.
• Equation of a plane in three-dimensional space typically uses the formula \(Ax + By + Cz = D\).

Through coordinate geometry, we can define planes based on their interaction with the coordinates. In our example, the point \( (3, 6, 12) \) lies on the plane, and using the knowledge that the plane is parallel to the \(xy\)-plane, aligned by its normal vector pointing along the \(z\)-axis, helps to set its explicit equation.
This approach makes finding the plane's equation straightforward and efficient by focusing on the z-values due to the plane's parallel nature. Understanding how planes interact spatially helps leverage geometrical concepts and algebra to resolve spatial problems.