Problem 73 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the distance from \((1,-1,2)\) to the plane \(x+3 y+z=7\).

Step-by-Step Solution

Answer

The distance from the point to the plane is \(\frac{7}{\sqrt{11}}\).

Step 1: Understanding the Problem

We need to find the shortest distance from the point \((1, -1, 2)\) to the plane given by the equation \(x + 3y + z = 7\). This can be calculated using the formula for the distance from a point to a plane.

Step 2: Identifying Plane Components

The plane equation is given in the form \(ax + by + cz = d\). Here, we identify \(a = 1\), \(b = 3\), \(c = 1\), and \(d = 7\).

Step 3: Identifying Point Coordinates

The coordinates of the point \((x_0, y_0, z_0)\) are given as \((1, -1, 2)\).

Step 4: Applying the Distance Formula

Use the formula for the distance from a point to a plane: \[ D = \frac{|ax_0 + by_0 + cz_0 - d|}{\sqrt{a^2 + b^2 + c^2}} \] to calculate the distance.

Step 5: Substituting Values into the Formula

Substitute the identified values into the formula: \(a = 1\), \(b = 3\), \(c = 1\), \(d = 7\), \((x_0, y_0, z_0) = (1, -1, 2)\). This gives: \[ D = \frac{|1 \cdot 1 + 3 \cdot (-1) + 1 \cdot 2 - 7|}{\sqrt{1^2 + 3^2 + 1^2}} \] which simplifies to \[ D = \frac{|1 - 3 + 2 - 7|}{\sqrt{1 + 9 + 1}} = \frac{| -7 |}{\sqrt{11}} \].

Step 6: Calculating the Distance

Simplify the expression. We have: \[ D = \frac{7}{\sqrt{11}} \]. This represents the distance from the point to the plane.

Key Concepts

Plane Equation

A plane in 3D space is often described by a linear equation: \(ax + by + cz = d\). This equation defines the set of all points \((x, y, z)\) that lie on the plane. Each of the coefficients \(a, b,\) and \(c\) in the equation play a crucial role: they form a normal vector to the plane. The term \(d\) is related to the distance from the origin to the plane along this normal vector.
Understanding the plane equation helps identify its orientation and positioning in space. Importantly, the normal vector \((a, b, c)\) is perpendicular to every line that lies entirely in the plane. This perpendicular nature is fundamental when calculating distances from points to the plane.

• **Normal Vector:** The vector \((a, b, c)\) that is perpendicular to the plane.
• **Constant \(d\):** Determines the exact position of the plane in relation to the origin.

Distance Formula

The distance formula for finding the shortest path from a point to a plane is: \[ D = \frac{|ax_0 + by_0 + cz_0 - d|}{\sqrt{a^2 + b^2 + c^2}} \]This formula derives from the geometric properties of planes in 3D space. The numerator \(|ax_0 + by_0 + cz_0 - d|\) calculates the signed perpendicular distance from the point to the plane. The absolute value ensures that the distance is positive.

• The denominator \(\sqrt{a^2 + b^2 + c^2}\) normalizes the distance, considering the normal vector's magnitude, allowing us to find the shortest possible distance.

To apply the distance formula effectively, substitute the appropriate values from the plane equation and the point's coordinates into the formula. Solving the resulting expression gives you the needed distance.

3D Coordinates

3D Coordinates help in locating positions in three-dimensional space using an ordered set of three numbers \((x, y, z)\). Each of these coordinates corresponds to values along the respective axes of a 3D Cartesian coordinate system.

• The \(x\)-coordinate denotes the position along the horizontal axis.
• The \(y\)-coordinate functions along the vertical axis.
  
• The \(z\)-coordinate specifies height or depth, differentiating 3D from the simpler 2D systems.

Understanding 3D coordinates is crucial when dealing with problems involving three-dimensional spaces, such as finding distances to planes. In our problem, the coordinates \((1, -1, 2)\) represent a specific point in this 3D space from which we want to measure the distance to a given plane.

Mathematics Problem-solving

Mathematics problem-solving involves structured steps to find solutions. In the context of finding the distance from a point to a plane, the process includes:

• **Understanding the Problem:** Begin by breaking down what is given and what needs to be found.
• **Identifying Necessary Elements:** Extract the coefficients from the plane equation and identify the coordinates of the point.
• **Applying Appropriate Formulas:** Use the distance formula accurately by carefully substituting values.

Problem-solving requires a blend of understanding theoretical concepts and applying practical calculation skills. Using formulas like the point-to-plane distance not only involves substituting numbers but also understanding each term's role in the calculation. Mastery comes from practicing with different examples, enhancing both mathematical intuition and confidence.