Problem 71 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the equation of the plane passing through \((-4,-1,2)\) and parallel to (a) the \(x y\)-plane (b) the plane \(2 x-3 y-4 z=0\)

Step-by-Step Solution

Answer

(a) The equation is \(z = 2\). (b) The equation is \(2x - 3y - 4z = -13\).

Step 1: Understanding the Equation of a Plane

Any plane can be defined by the equation \(Ax + By + Cz = D\), where \(A, B, C\) are the coefficients that form the normal vector to the plane, and \(D\) is a constant. The normal vector \((A, B, C)\) is perpendicular to the plane, which means any vector parallel to the plane should have dot product zero with this normal vector.

Step 2: Equation for Plane Parallel to the xy-plane

A plane parallel to the \(xy\)-plane has a normal vector that is parallel to the \(z\)-axis. Hence, the normal vector can be represented as \((0, 0, 1)\). Therefore, the equation of the plane takes the form \(0x + 0y + 1z = D\), simplifying to \(z = D\). To find \(D\), use the point \((-4,-1,2)\), giving \(z = 2\). Thus, the equation is \(z = 2\).

Step 3: Finding Equation for Plane Parallel to another Plane

For the plane parallel to \(2x - 3y - 4z = 0\), their normal vectors must be identical to maintain parallelism. Hence, our plane will also have the normal vector \((2, -3, -4)\). Substitute the given point \((-4,-1,2)\) into \(2x - 3y - 4z = D\) to find \(D\):\[2(-4) - 3(-1) - 4(2) = D\] Evaluating, \(D = -8 + 3 - 8 = -13\). The required plane equation is \(2x - 3y - 4z = -13\).

Key Concepts

Normal Vector

When dealing with equations of planes, the concept of a normal vector is vital. A normal vector is a vector that is perpendicular to a plane. This means it forms a right angle with any vector that lies on the plane itself.
In mathematical terms, if a plane is represented by the equation \(Ax + By + Cz = D\), the vector \((A, B, C)\) serves as its normal vector.

• This vector holds an essential role in identifying the orientation of a plane in three-dimensional space.
• The components \(A\), \(B\), and \(C\) are direction ratios for the line perpendicular to the plane.

The normal vector also enables us to determine if two planes are parallel, since parallel planes share the same normal vector.

Parallel Planes

To understand parallel planes, let's start with what it means for two planes to be parallel. Two planes are parallel if they do not intersect, and they are aligned in the same direction.
The key to determining if two planes are parallel lies in comparing their normal vectors. If the normal vectors are the same or scalar multiples of each other, then the planes are parallel.

• For example, a plane parallel to the \(xy\)-plane will have a normal vector that aligns with the \(z\)-axis, such as \((0, 0, 1)\).
• Meanwhile, a plane parallel to another plane like \(2x - 3y - 4z = 0\) will have an identical normal vector \((2, -3, -4)\).

Understanding this helps in formulating the equation of a desired plane that is parallel to a given one.
You adjust the constant \(D\) while keeping the coefficients \(A, B, C\) (from the normal vector) consistent.

Coordinates

Coordinates are used to specify a point in space and are crucial in formulating the equation of a plane. A point is represented in three-dimensional space with coordinates \((x, y, z)\).

• These values indicate the point's exact location on the Cartesian plane, which consists of the \(x\), \(y\), and \(z\) axes.
• For any plane described by the equation \(Ax + By + Cz = D\), plugging in the coordinates of a point known to lie on the plane will help solve for the constant \(D\).

In the context of our exercise, the point \((-4, -1, 2)\) was used to determine \(D\) for the planes parallel to both the \(xy\)-plane and another given plane. This involves substituting \((-4, -1, 2)\) into plane equations and solving for the unknown value.
Coordinates make it easy to anchor a plane in space, giving you a fixed reference point.

Dot Product

The dot product is a way to multiply two vectors, resulting in a scalar (a single number). It plays an important role in determining relationships between vectors, especially with planes.
The dot product between two vectors \((a_1, a_2, a_3)\) and \((b_1, b_2, b_3)\) is calculated as \(a_1 b_1 + a_2 b_2 + a_3 b_3\).

• When the dot product of a vector parallel to the plane and the plane's normal vector equals zero, this indicates perpendicularity.
• This property is used to confirm that vectors are indeed lying on the plane or parallel to the plane when its dot product with the normal vector is zero.

In finding plane equations, understanding the concept of the dot product helps ensure that desired orientations and parallel conditions are met. For example, ensuring a plane's equation consists correctly of its normal vector components as seen in our exercises.