Question

Sketch several representative vectors in the vector field. $$ \mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}+z \mathbf{k} $$

Short answer

At point (x, y, z) the vector points in the direction of increase of x, y and z coordinates respectively. The magnitude equals the distance from the origin. Therefore, sketch arrows in the +x, +y, +z directions at points (1,0,0), (0,1,0), (0,0,1), and in the -x, -y directions at points (-1,0,0), (0,-1,0). Repeat this for other selected points as well.

Step-by-step solution

Understand what the Vector Field represents

The given Vector Field \( \mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}+z \mathbf{k} \) represents a field where at each point (x, y, z), there is a vector which points in the direction of the increase of x, y and z coordinates respectively. The magnitude (or size) of the vector equals the distance from the origin.

Select representative points

Select several points in the three-dimensional space, let's say five here for simplicity: (1,0,0), (0,1,0), (0,0,1), (-1, 0, 0), and (0, -1, 0).

Determine and visualize the vectors at chosen points

Evaluate \( \mathbf{F} \) at the chosen points: \n(1,0,0) yields the vector \( \mathbf{i} \), pointing in the +x direction with magnitude 1. Hence, at point (1,0,0), plot \( \mathbf{i} \) direction with an arrow. \nSimilarly, for the remaining points:\n(0,1,0) yields \( \mathbf{j} \), so we plot an arrow in the +y direction at this point.\n(0,0,1) yields \( \mathbf{k} \), hence an arrow in the +z direction is plotted at this point.\n(-1,0,0) yields \( -\mathbf{i} \) indicating a vector in the -x direction, plot it at this point.\n(0,-1,0) yields \( -\mathbf{j} \), hence an arrow in the -y direction is plotted at this point.

Repeat the process

Repeat the process to fill the space with representative vectors.

Key concepts

Vector Field Visualization

Visualizing a vector field involves creating a graphical representation to illustrate how vectors are organized in a specified space. For our exercise with the vector field \( \mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}+z \mathbf{k} \), imagine the three-dimensional space filled with arrows. Each arrow starts at a point \((x, y, z)\) and points in a certain direction, with a length corresponding to the vector's magnitude at that point. To simplify the illustration, we typically choose a selection of points and draw the vectors at just those points. The goal is to provide a snapshot that gives us insight into the behavior of the vector field throughout the space.

When visualizing a vector field, it is crucial to remember that the vectors represent both a direction and a magnitude, which are equally vital. For the given vector field, visualizing it will reveal that as we move away from the origin, the vectors increase in length. This indicates that the magnitude of the vector field is directly proportional to the distance from the origin. By strategically choosing points and drawing corresponding vectors, you get a clearer understanding of the entire field's properties without drawing every single vector.

Vector Field Representation

To represent a vector field effectively, we use mathematical notation and diagrams. The given vector field \( \mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}+z \mathbf{k} \) demonstrates this. The expression tells us that at any point \((x, y, z)\), the vector field assigns a vector consisting of three components, each corresponding to the standard unit vectors \(\mathbf{i}, \mathbf{j}, and \mathbf{k}\) in three-dimensional space.

When choosing how to represent this vector field, one can use arrows of different lengths and directions, plotted at various points selected from the space. A good representation will also consider the scale, ensuring that the vectors are not too long or too short, which could mislead the observer about the field's magnitude variations. Additionally, the direction of the arrows should accurately indicate the vector direction at each point. Combining these representations with the mathematical form of the vector field allows for a comprehensive understanding of the field's characteristics.

Three-Dimensional Vectors

Three-dimensional vectors are mathematical objects with three components, corresponding to three mutually perpendicular directions typically denoted as the \(x\), \(y\), and \(z\)-axes in a Cartesian coordinate system. They can represent physical quantities like force, velocity, or magnetic field, among others. In the context of our exercise with vector field \( \mathbf{F}(x, y, z)\), each vector's three components are projections along the \(x\), \(y\), and \(z\)-axes.

Three-dimensional vectors can be visualized as arrows starting at the origin (or another initial point) and pointing in space. The length of the arrow corresponds to the vector's magnitude, and the direction in which the arrow points signifies the vector's direction. To represent vectors like \(\mathbf{i}, \mathbf{j}, and \mathbf{k}\), we use arrows pointing along the positive \(x\), \(y\), and \(z\)-axes with a length of 1, as these are unit vectors. The concept of three-dimensional vectors is fundamental to understanding vector fields, as they provide the structure to describe how the field behaves in space.