Problem 87 - Calculus Early Transcendentals: Pearson New International Edition

Question

Suppose that the three coordinate planes bounding the first octant are mirrors. A light ray with direction \(a \mathbf{i}+b \mathbf{j}+c \mathbf{k}\) is reflected successively from the \(x y\)-plane, the \(x z\)-plane, and the \(y z\)-plane. Determine the direction of the ray after each reflection, and state a nice conclusion concerning the final reflected ray.

Step-by-Step Solution

Answer

The final direction of the ray is \(-a \mathbf{i} - b \mathbf{j} - c \mathbf{k}\), a reversal of the original direction.

Step 1: Understand the Problem Context

We have a light ray represented by a vector \( \mathbf{v} = a \mathbf{i} + b \mathbf{j} + c \mathbf{k} \) traveling in the first octant and reflecting off three coordinate planes: the \(xy\)-plane, \(xz\)-plane, and \(yz\)-plane. We need to determine how the direction of the vector changes after each reflection.

Step 2: Reflect from the xy-plane

When a light ray hits the \(xy\)-plane, the \(z\)-component of the vector is inverted, while \(x\) and \(y\) components remain unchanged. Thus, after this reflection, the vector becomes \( a \mathbf{i} + b \mathbf{j} - c \mathbf{k} \).

Step 3: Reflect from the xz-plane

The next reflection is off the \(xz\)-plane, where the \(y\)-component of the vector is inverted. The vector now becomes \( a \mathbf{i} - b \mathbf{j} - c \mathbf{k} \).

Step 4: Reflect from the yz-plane

Finally, the ray reflects off the \(yz\)-plane, inverting the \(x\)-component. Therefore, the vector becomes \( -a \mathbf{i} - b \mathbf{j} - c \mathbf{k} \).

Step 5: Conclusion

The final direction of the reflected light ray is the negative of the initial direction. So the final outcome is that the ray's direction is reversed compared to the original direction \(\mathbf{v} = a \mathbf{i} + b \mathbf{j} + c \mathbf{k}\).

Key Concepts

Coordinate Geometry

Coordinate geometry is a crucial mathematical tool that helps us understand the position and movement of objects in space. In the context of vector reflection, coordinate geometry provides a framework for tracking how vectors change direction upon interacting with surfaces. When dealing with reflections, we often consider the planes in a three-dimensional coordinate system: the xy-plane, the xz-plane, and the yz-plane.

This system uses the Cartesian coordinates (x, y, z) to describe any point or vector in space. Each plane is defined by holding one of these coordinates constant, which allows us to visualize reflections:

• The xy-plane is where the z-coordinate is zero.
• The yz-plane is where the x-coordinate is zero.
• The xz-plane is where the y-coordinate is zero.

Understanding how vectors behave relative to these planes is essential for analyzing and predicting the results of interactions like reflections.

Vector Components

Vectors have multiple components that play into their behavior in space. Specifically, they can be broken down into components aligned with each axis in a 3D space, represented as \( \mathbf{v} = a \mathbf{i} + b \mathbf{j} + c \mathbf{k} \), where:

• \( a \) is the x-component aligning with the x-axis.
• \( b \) is the y-component aligning with the y-axis.
• \( c \) is the z-component aligning with the z-axis.

Each component indicates how much influence the vector has along that particular axis. When a vector reflects off a plane, the component perpendicular to that plane changes its sign.

For example, when reflecting off the xy-plane, the z-component (\( c \)) flips from positive to negative, as shown from \( \mathbf{v} = a \mathbf{i} + b \mathbf{j} + c \mathbf{k} \rightarrow a \mathbf{i} + b \mathbf{j} - c \mathbf{k} \). This flipping behavior illustrates how specific vector components transform with each reflection.

Physics of Light Reflections

The physics of light reflections can be well-understood through vector mathematics in coordinate geometry. When light hits a surface, it reflects according to specific laws which depend on the nature of the surface and the angle of incidence. These reflections can be described using vectors.

In a perfect reflection scenario, such as bouncing off the planes of an octant in a coordinate system, the behavior is predictable and systematic. Each reflection inverts the component of the vector that is perpendicular to the reflecting plane. Light does not lose any components parallel to the plane, so their direction remains stable.

• First reflection off xy-plane: Again, inverts the z-component.
• Second reflection off xz-plane: Inverts the y-component.
• Third reflection off yz-plane: Inverts the x-component.

The result of this sequence of reflections is the reversal of the light ray's direction, showing the consistency of vector reflection in physics.

Mathematical Problem Solving

Solving problems involving vector reflection requires a solid strategy and understanding of the underlying mathematics. The steps to tackle such an exercise often involve:

• Understanding the geometry and coordinate system in which the vectors exist.
• Identifying how each component of the vector changes with each reflection.
• Determining the sequence of modifications through logical deductions.

Here, we follow the vector as it reflects off the xy, xz, and yz planes, observing how each perpendicular component inverts upon striking a plane. By systematically analyzing each step, students can logically deduce that the vector's final direction is the exact opposite of its original direction, \( -a \mathbf{i} - b \mathbf{j} - c \mathbf{k} \). This conclusion illustrates how mathematical problem solving can effectively simplify and solve complex physical phenomena.