Question

Find the equation of the plane through \((-1,2,-3)\) and parallel to the plane \(2 x+4 y-z=6\).

Short answer

The equation of the plane is \(2x + 4y - z = 9\).

Step-by-step solution

Understand the problem

We need to find the equation of a plane that goes through a certain point and is parallel to a given plane. The given plane's equation is \(2x + 4y - z = 6\), and we want our new plane to go through the point \((-1, 2, -3)\).

Determine the normal vector

Parallel planes have the same normal vector. The normal vector of the given plane, \(2x + 4y - z = 6\), is \(\langle 2, 4, -1 \rangle\). Hence, the plane we are looking for has the same normal vector \(\langle 2, 4, -1 \rangle\).

Use the point-normal form of a plane

The equation of a plane with a normal vector \(\langle A, B, C \rangle\) passing through a point \((x_0, y_0, z_0)\) is \(A(x - x_0) + B(y - y_0) + C(z - z_0) = 0\). Substitute \(A = 2\), \(B = 4\), \(C = -1\), and point \((-1, 2, -3)\) to get the equation.

Substitute into the point-normal form

Substitute the values into the point-normal form: \[2(x + 1) + 4(y - 2) - 1(z + 3) = 0\].

Simplify the equation

Expand and simplify the equation: \[2x + 2 + 4y - 8 - z - 3 = 0\], which simplifies to \[2x + 4y - z - 9 = 0\].

Write the final equation

Thus, the simplified equation of the plane is \(2x + 4y - z = 9\).

Key concepts

Vector Geometry

Vector geometry is a powerful tool used to describe geometric objects like lines and planes in space. It's a branch of mathematics that helps us easily solve problems involving positions and directions.
Picturing vectors as arrows, they have both magnitude and direction. In our current context, vectors are essential in defining the normal to a plane or line.
Understanding and using vectors significantly simplify the description and calculations of geometric problems. They allow us to develop equations of planes, lines, and even complex surfaces using operations like addition, subtraction, and scalar multiplication.

• Vectors can represent points in space such as the position vector \((x_0, y_0, z_0)\).
• They can also represent directions, like the normal vector of a plane.
• Operations like the dot product come into play when we need to define angles and orthogonality.

This foundational knowledge of vector geometry supports the entire process of finding plane equations.

Point-Normal Form

The point-normal form of a plane equation is a particularly direct method to describe a plane in space.
This form utilizes a normal vector to the plane and a point through which the plane passes.
The general equation is given by:\[A(x - x_0) + B(y - y_0) + C(z - z_0) = 0\]Here, \(A, B, C\) are the components of the normal vector, and \(x_0, y_0, z_0\) are the coordinates of the point on the plane.

• Choosing the correct normal vector is crucial: it defines the orientation of the plane.
• Substituting the known values into the equation gives a concrete form of our desired plane.

Using this equation effectively allows us to find the equation of a plane quickly when a point and a normal vector are provided.

Parallel Planes

Parallel planes are an interesting concept in geometry characterized by having the same orientation in space.
Planes are parallel if their normal vectors are equal or scalar multiples of each other.
This property makes it easier to identify parallel planes and solve related problems.

• For given planes with equations in the form \(Ax + By + Cz = D\), if normal vectors such as \(\) are the same, the planes are parallel.
• In our example, both planes have the normal vector \(\langle 2, 4, -1 \rangle\), confirming their parallel nature.

This understanding simplifies finding new plane equations that are parallel to known ones, as only the constant term, derived from the specific point on the new plane, changes.