Question

Three-dimensional vector fields Sketch a few representative vectors of the following vector fields. $$\mathbf{F}=\frac{\langle x, y, z\rangle}{\sqrt{x^{2}+y^{2}+z^{2}}}$$

Short answer

Based on the given vector field $$\mathbf{F}=\frac{\langle x, y, z\rangle}{\sqrt{x^{2}+y^{2}+z^{2}}},$$ describe the behavior of the vector field in different regions of the three-dimensional space.

Step-by-step solution

Calculate representative vectors

First, we need to calculate the representative vectors of the vector field at different points in the space. We can do this by plugging in values for x, y, and z into the equation of the vector field, and then simplifying the expression.

Plot representative vectors along x-axis

We will calculate the vector field at points (±1, 0, 0), (±2, 0, 0), etc., and plot them. For example, at point (1,0,0), $$\mathbf{F}(1, 0, 0) = \frac{\langle 1, 0, 0 \rangle}{\sqrt{1^2+0^2+0^2}} = \langle 1, 0, 0 \rangle$$

Plot representative vectors along y-axis

We will calculate the vector field at points (0, ±1, 0), (0, ±2, 0), etc., and plot them. For example, at point (0,1,0), $$\mathbf{F}(0, 1, 0) = \frac{\langle 0, 1, 0 \rangle}{\sqrt{0^2+1^2+0^2}} = \langle 0, 1, 0 \rangle$$

Plot representative vectors along z-axis

We will calculate the vector field at points (0, 0, ±1), (0, 0, ±2), etc., and plot them. For example, at point (0,0,1), $$\mathbf{F}(0, 0, 1) = \frac{\langle 0, 0, 1 \rangle}{\sqrt{0^2+0^2+1^2}} = \langle 0, 0, 1 \rangle$$

Plot representative vectors around the origin

We can calculate the vector field at points such as (1,1,1), (1,-1,-1), etc., considering at least one point from each octant and plot them. For example, at point (1,1,1), $$\mathbf{F}(1, 1, 1) = \frac{\langle 1, 1, 1 \rangle}{\sqrt{1^2+1^2+1^2}} = \frac{\langle 1, 1, 1 \rangle}{\sqrt{3}}$$ After plotting these representative vectors, we can notice the behavior of the vector field in different regions of the three-dimensional space.

Key concepts

Three-dimensional Space

Three-dimensional space is an essential concept in understanding vector fields like \( \mathbf{F} = \frac{\langle x, y, z\rangle}{\sqrt{x^{2}+y^{2}+z^{2}}} \). It consists of three mutually perpendicular axes: the x-axis, y-axis, and z-axis. Each point in this space can be represented by three coordinates \((x, y, z)\).

This space allows us to describe positions, compute distances, and analyze directions in a more vivid and dynamic manner compared to two-dimensional space. For example, the distance from the origin \((0,0,0)\) to any point \((x, y, z)\) is calculated using the formula \(\sqrt{x^2 + y^2 + z^2}\), which can help define the magnitude of vectors in this space.

The importance of three-dimensional space lies in its applications, such as:

• Physics, where objects are analyzed in a three-dimensional framework.
• Engineering, for designing structures and systems that have depth.
• Computer graphics, to render realistic environments and animations.

Understanding this concept is pivotal for grasping the behavior of vector fields and their representations in three dimensions.

Vector Calculus

Vector calculus is an advanced mathematical tool that helps analyze vector fields, differentiating and integrating vectors within three-dimensional space. It extends the principles of calculus to vectors, dealing with operations such as calculating divergence, curl, and flux.

In the given exercise, the vector field \(\mathbf{F} = \frac{\langle x, y, z\rangle}{\sqrt{x^{2}+y^{2}+z^{2}}} \) is a classic example of a vector function, where each point \((x, y, z)\) in space is associated with a vector. The expression \(\sqrt{x^2 + y^2 + z^2}\) is instrumental in determining the vector's magnitude, thus normalizing it.

Vector calculus enables us to:

• Study how vector fields change over space.
• Understand the flow of fields, crucial for fields like electromagnetism.
• Solve real-world problems involving fields, forces, and fluids.

Engaging with vector calculus is crucial for understanding not only mathematical theory but also its extensive application in science and engineering fields.

Vector Components

Vector components are fundamental in breaking down a vector into its basic parts along the coordinate axes. For a vector \(\langle x, y, z \rangle\), its components are the projections onto the x, y, and z-axes.

Considering the vector field \(\mathbf{F} = \frac{\langle x, y, z\rangle}{\sqrt{x^{2}+y^{2}+z^{2}}} \), the components \(x, y, z\) are essential for understanding the direction and magnitude of vectors at any point in space. These components allow the vector field to be expressed and visualized at various points such as \((1,0,0)\), \((0,1,0)\), and \((0,0,1)\).

Understanding vector components helps in:

• Finding resultant vectors by adding individual components.
• Applying transformations and rotations in three-dimensional space.
• Simplifying complex vector calculations by working with smaller parts.

Grasping the concept of vector components is key to effectively sketching and analyzing vector fields in three-dimensional space.