Question

Find the equation of the plane through \((2,5,1)\) that is parallel to the plane \(x-y+2 z=4\).

Short answer

The equation of the plane is \(x - y + 2z = -1\).

Step-by-step solution

Understand the Given Plane Equation

The given plane equation is \(x - y + 2z = 4\). This equation is in the standard form \(Ax + By + Cz = D\), where \(A=1\), \(B=-1\), and \(C=2\). The normal vector to this plane is \(\vec{n} = (1, -1, 2)\).

Determine Normal Vector for the New Plane

Since the new plane is parallel to the given plane, it shares the same normal vector. Therefore, the normal vector to the new plane is \(\vec{n} = (1, -1, 2)\).

Use the Normal Vector and Point to Write the Plane Equation

Use the point \((x_0, y_0, z_0) = (2, 5, 1)\) that lies on the new plane. The equation of a plane with a normal vector \(\vec{n} = (A, B, C)\) through a point \((x_0, y_0, z_0)\) is given by \(A(x-x_0) + B(y-y_0) + C(z-z_0) = 0\).

Substitute Values into the Plane Equation

Substitute the values \(A = 1\), \(B = -1\), \(C = 2\), and the point \((2, 5, 1)\) into the equation: \[1(x-2) - 1(y-5) + 2(z-1) = 0\] This simplifies to: \[x - 2 - y + 5 + 2z - 2 = 0\]

Simplify the Plane Equation

Combine like terms to simplify: \[x - y + 2z + 1 = 0\] Thus, the equation of the required plane is \(x - y + 2z = -1\).

Key concepts

Normal Vector

The concept of a normal vector is essential when dealing with planes in geometry. A normal vector to a plane is a vector that is perpendicular to every line lying on the plane. This simple property makes it a powerful tool because it defines the plane's orientation in space.

• Every plane in 3D geometry can be represented by its normal vector. This vector dictates the tilt and direction of the plane.
• For the plane equation in the form \(Ax + By + Cz = D\), the components \((A, B, C)\) constitute the normal vector.
• In our example, the given plane \(x - y + 2z = 4\) has a normal vector \((1, -1, 2)\), which is derived directly from the coefficients of \(x\), \(y\), and \(z\).

Being aware of the normal vector allows us to quickly determine relationships between multiple planes, such as parallelism.

Parallel Planes

Planes are parallel when their normal vectors are identical or scalar multiples of each other. This means they share the same direction or are aligned in space in a way that they never intersect.

• If two planes have the same normal vector, they are parallel. For instance, both \(x - y + 2z = 4\) and \(x - y + 2z = -1\) have the normal vector \((1, -1, 2)\).
• Since the normal vectors are identical, these planes remain equidistant from each other, like layers on a cake, extending endlessly without touching.

To find an equation for a plane parallel to another and passing through a specific point, we use the same normal vector and adjust only for the new point of passage.

Standard Form Equation

A plane in three-dimensional space can commonly be expressed using the standard form equation: \(Ax + By + Cz = D\). This equation simplifies the representation of a plane by using linear combinations of the variables \(x\), \(y\), and \(z\).

• Standard form makes it easy to identify the normal vector, shown by the coefficients \(A, B, C\).
• Knowing the standard form allows us to write the equation for any parallel plane. In this specific task, we determine the equation for a plane parallel to \(x - y + 2z = 4\), passing through the point \((2, 5, 1)\).
• We use the formula \(A(x-x_0) + B(y-y_0) + C(z-z_0) = 0\) to form the new plane equation, substituting the new point \((2, 5, 1)\) and the identical normal vector \((1, -1, 2)\).

This approach not only provides the equation of the new plane \(x - y + 2z = -1\), but also showcases the utility of standard form in planar geometry.