Question

Three-dimensional vector fields Sketch a few representative vectors of the following vector fields. $$\mathbf{F}=\langle 1,0, z\rangle$$

Short answer

Based on the vector field $$\mathbf{F} = \langle 1, 0, z \rangle$$, sketch the representative vectors for the following points: 1. Point \((0,0,0)\): Vector \(\langle 1, 0, 0\rangle\) 2. Point \((1,0,1)\): Vector \(\langle 1, 0, 1\rangle\) 3. Point \((1,0,2)\): Vector \(\langle 1, 0, 2\rangle\) 4. Point \((2,0,-1)\): Vector \(\langle 1, 0, -1\rangle\) 5. Point \((2,0,0)\): Vector \(\langle 1, 0, 0\rangle\) 6. Point \((2,0,1)\): Vector \(\langle 1, 0, 1\rangle\) 7. Point \((3,0,-2)\): Vector \(\langle 1, 0, -2\rangle\) 8. Point \((3,0,-1)\): Vector \(\langle 1, 0, -1\rangle\) 9. Point \((3,0,0)\): Vector \(\langle 1, 0, 0\rangle\) Describe how the vector field behaves as the points move in the 3D space.

Step-by-step solution

Understand the Vector Field Formula

The vector field is given by $$\mathbf{F} = \langle 1, 0, z \rangle$$, which means that the \(x\) component of the vector is always 1, the \(y\) component is always 0, and the \(z\) component depends on the \(z\)-coordinate of the point in the 3D space.

Choose Points to Sketch Vectors

For a good representation of the vector field, we'll choose a variety of points with different \(z\)-coordinates. We'll choose points in the \(xz\)-plane since the \(y\)-component is always zero. Let's choose the following points: $$(0,0,0),\ (1,0,1),\ (1,0,2),\ (2,0,-1),\ (2,0,0),\ (2,0,1),\ (3,0,-2),\ (3,0,-1),\ (3,0,0)$$

Calculate the Vectors for Each Point

Now we'll calculate the vectors for each chosen point using the vector field formula:$$\mathbf{F} = \langle 1, 0, z \rangle$$ Here are the vectors for our chosen points: Point \((0,0,0)\): Vector \(\langle 1, 0, 0\rangle\) Point \((1,0,1)\): Vector \(\langle 1, 0, 1\rangle\) Point \((1,0,2)\): Vector \(\langle 1, 0, 2\rangle\) Point \((2,0,-1)\): Vector \(\langle 1, 0, -1\rangle\) Point \((2,0,0)\): Vector \(\langle 1, 0, 0\rangle\) Point \((2,0,1)\): Vector \(\langle 1, 0, 1\rangle\) Point \((3,0,-2)\): Vector \(\langle 1, 0, -2\rangle\) Point \((3,0,-1)\): Vector \(\langle 1, 0, -1\rangle\) Point \((3,0,0)\): Vector \(\langle 1, 0, 0\rangle\)

Sketch the Representative Vectors

Finally, we'll sketch the representative vectors with their tails at each point in 3D space. Each vector will have its \(x\)-component always pointing 1 unit in the positive \(x\)-direction and its \(y\)-component will be 0. The \(z\)-component of the vectors will depend on the point and will change according to the height \((x,y,z)\) in relation to the origin. Keep in mind that the vectors will grow or shrink in the \(z\)-direction depending on the \(z\)-coordinate of their tail. When the tail is above the \(xz\)-plane, the positive \(z\)-component vectors will point upwards, and when the tail is below the \(xz\)-plane, the negative \(z\)-component vectors will point downwards. When the tail is on the \(xz\)-plane, the \(z\)-component will be zero, and the vector will be purely in the \(x\)-direction. After sketching the vectors for the chosen points, you should see an overall pattern in the vector field indicating how the field behaves as the points move in the 3D space.

Key concepts

Vector components

In a three-dimensional vector field, vectors are often expressed in terms of their components. Each vector typically consists of three components corresponding to the three axes – x, y, and z. In this exercise, the vector is given by the formula \(\mathbf{F} = \langle 1, 0, z \rangle\). This means that every vector in this field has a fixed x-component value of 1, a y-component fixed at 0, and a z-component that varies based on the z-coordinate of the point at which the vector is evaluated.

Understanding vector components is crucial because they define the direction and magnitude of the vector. The x-component of 1 indicates that there will always be movement or force in the positive x-direction for this vector. The absence of a y-component (y = 0) means that there is no movement in the y-direction, which simplifies the visualization to two dimensions: x and z. Finally, the z-component varies, meaning it will differ at each point, thus affecting the vector's behavior along the z-axis.

This concept of vector components helps us to understand how vectors represent multivariate forces or directions within a three-dimensional space, making it an essential tool in fields like physics and engineering.

3D coordinate system

A three-dimensional coordinate system is a framework that enables us to specify the location of points in space through three numerical coordinates. The system is constituted by three perpendicular axes labeled as x, y, and z. Their intersection point is known as the origin, denoted by (0, 0, 0).

In the context of this exercise, the vectors are influenced primarily by changes in the z-coordinate. The x-axis influences vectors consistently since the x-component is always 1. However, the y-axis does not influence the vectors as the y-component is always 0, indicating no variation along that axis.

• The x-axis moves horizontally from left to right.
• The y-axis moves vertically up and down.
• The z-axis moves perpendicularly to both, extending in and out of the plane formed by x and y.

Recognizing how these axes intersect and define space helps us plot the vectors accurately and understand the spatial distribution and variation of the vector field in a three-dimensional framework.

Sketching vectors

Sketching vectors in a 3D space can seem daunting, but by breaking it down, it becomes manageable. The vectors are represented by arrows, where the starting point (tail) is at a specific location, and the arrowhead points to the vector's end, indicating direction.

To sketch vectors for the field \(\mathbf{F} = \langle 1, 0, z \rangle\), focus on these steps:

• Choose a variety of points, as seen in the provided list of coordinates, spread across the xz-plane (since y=0 eliminates the y-axis effect).
• Calculate the corresponding vector for each point using the formula. Note how the z-component changes with the z-coordinate of each point.
• Draw each vector with its tail at the given point, ensuring the x-component always extends one unit in the positive x-direction.
• Observe the z-component; longer vectors indicate a large z-value and thus extend further in their corresponding direction up (positive z) or down (negative z).

This method helps in visualizing the entire vector field, revealing patterns and behaviors as vectors vary based on their position. An overall glance would show vectors with consistent x movement, no y movement, and variable z movement, reflecting the field's unique characteristics.