Problem 67 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 65-68, find the equation of the plane having the given normal vector \(\mathbf{n}\) and passing through the given point \(P\). $$ \mathbf{n}=\langle 1,4,4\rangle ; P(1,2,1) $$

Step-by-Step Solution

Answer

The equation of the plane is \( x + 4y + 4z = 13 \).

Step 1: Understanding the Equation of a Plane

To find the equation of a plane given a normal vector \( \mathbf{n} = \langle a, b, c \rangle \) and a point \( P(x_0, y_0, z_0) \) on the plane, we use the formula: \[ a(x - x_0) + b(y - y_0) + c(z - z_0) = 0 \] In this case, \( a = 1 \), \( b = 4 \), \( c = 4 \), and the point \( P = (1, 2, 1) \).

Step 2: Substituting Values into the Formula

Substitute the values of \( a \), \( b \), \( c \), and the coordinates of the point \( P \, (x_0=1, y_0=2, z_0=1) \) into the formula: \[ 1(x - 1) + 4(y - 2) + 4(z - 1) = 0 \]

Step 3: Simplifying the Equation

Simplify each term in the equation: \[ x - 1 + 4y - 8 + 4z - 4 = 0 \] Combine like terms: \[ x + 4y + 4z - 13 = 0 \]

Step 4: Final Equation of the Plane

Reorder and present the final equation of the plane: \[ x + 4y + 4z = 13 \] This is the equation of the plane with the given normal vector and point.

Key Concepts

Normal Vector

A normal vector is crucial when discussing the equation of a plane. When we describe a plane in three-dimensional space, this vector is perpendicular to the plane's surface. Consider it as an arrow pointing straight out of a flat surface.
For instance, given a normal vector \( \mathbf{n} = \langle 1, 4, 4 \rangle \), each component can be seen as coefficients impacting the direction and steepness of the plane. These numbers dictate the unique orientation of the plane in space.

• Significance of Components: Each component corresponds to the direction in x, y, and z axes.
• Interaction with Plane: A change in these values results in a rotation of the plane.

The vector's orientation helps determine how steeply the plane approaches these axes, influencing its entire position in the 3D space.

Point on a Plane

A point on a plane acts as an anchor, providing a specific location through which the plane passes. In our context, this is point \( P(1, 2, 1) \).
Having a fixed point helps us to uniquely position the plane in conjunction with its normal vector. This is because:

• Reference Point: It ensures that the plane isn't just floating in space without a defined position.
• Determining the Equation: Once a point is set, substituting it into the plane equation allows us to solve for unknowns, ensuring a tangible placement of the plane.

For instance, through point \( P \), we substitute \((x_0, y_0, z_0) = (1, 2, 1)\) in the equation, anchoring our plane effectively in the 3D space.

Cartesian Equation

The Cartesian equation is a straightforward method to describe planes in three-dimensional space using linear algebra. It's in the form: \[ a(x - x_0) + b(y - y_0) + c(z - z_0) = 0 \] This captures the essence of the plane's geometry.

• Coefficient Role: The coefficients \( a, b, \) and \( c \) are derived from the normal vector \( \mathbf{n} \).
• Point of concern: \((x_0, y_0, z_0) \) are coordinates of the known point \( P \) on the plane.

In our scenario, using the given normal \( \langle 1, 4, 4 \rangle \) and point \( P(1, 2, 1) \), we substitute them into the formula. Simplifying such an equation stabilizes the plane's description in numerical terms, giving us: \[ x + 4y + 4z = 13 \] This represents the complete, easily usable linear form of our plane's equation.

Vector Calculus

Vector calculus is the mathematical study that deals with vector fields and their derivatives and integrals. It's essential when working with dimensions beyond the simple line or curve, as it involves complex geometry like planes.
In context, when finding a plane's equation:

• Use of Derivatives: Vector calculus helps in determining the rise and fall of surfaces i.e., planes, based on their vectors.
• Field Interactions: It describes how planes and vectors interact, especially relevant in physics and engineering where forces and fields are spatially determined.

Here, using a normal vector, the direction and steepness of the plane were quantified via calculus principles to derive its equation. The Cartesian equation then becomes a practical application of these vector calculus operations, tying theoretical math to practical geometric problems.