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Chapter 17: Problem 37

Let \(\overrightarrow{\boldsymbol{F}}=x \vec{i}+y \vec{j}\) and \(\overrightarrow{\boldsymbol{G}}=-y \vec{i}+x \vec{j}\) Sketch the vector field \(\vec{L}=\vec{F}+b \vec{G}\) if: (a) \(\quad b=0\) (b) \(\quad b>0\) (c) \(\quad b<0\)

Short Answer

Expert verified
For the vector field \( \vec{L} = \vec{F} + b \vec{G} \): (a) When \( b = 0 \), vectors are radial. (b) When \( b > 0 \), vectors rotate counter-clockwise. (c) When \( b < 0 \), vectors rotate clockwise.

Step by step solution

01

Understanding the Given Vectors

The vector field \( \overrightarrow{\boldsymbol{F}} = x \vec{i} + y \vec{j} \) represents a vector field pointing in the direction of increasing \( x \) and \( y \). Meanwhile, \( \overrightarrow{\boldsymbol{G}} = -y \vec{i} + x \vec{j} \) represents a vector field that rotates 90 degrees counter-clockwise from \( \overrightarrow{\boldsymbol{F}} \).
02

Defining the Vector Field for Part (a)

In this case, we need to find \( \vec{L} = \vec{F} + b \vec{G} \) when \( b = 0 \). Substitute \( b = 0 \) to get \( \vec{L} = \vec{F} \). Thus, \( \vec{L} = x \vec{i} + y \vec{j} \). This vector field is pointing away from the origin in the direction of each point \( (x, y) \), representing a radial field.
03

Analyzing the Vector Field for Part (b)

For \( b > 0 \), each vector in \( \vec{L} = \vec{F} + b \vec{G} \) will have a component along an anti-clockwise direction due to \( \vec{G} \). At point \((x, y)\), \( \vec{L} = (x - by) \vec{i} + (y + bx) \vec{j} \). As \( b \) increases, the rotation impact intensifies.
04

Exploring the Vector Field for Part (c)

For \( b < 0 \), we observe that the component from \( \vec{G} \) is now being subtracted, leading each vector in \( \vec{L} = \vec{F} + b \vec{G} \) to potentially rotate clockwise. At point \((x, y)\), the vector becomes \( (x - by) \vec{i} + (y + bx) \vec{j} \). As \( b \) becomes more negative, the clockwise rotation effect becomes more prominent.
05

Sketching the Vector Fields

To sketch: - For \( b=0 \), vectors radiate outward from the origin. - For \( b>0 \), the vectors in the field curve anti-clockwise around the origin since they have positive values of \( b \). - For \( b<0 \), the field exhibits clockwise curvature since the vectors include the negative rotation component associated with \( b \). Draw vectors at various \( x, y \) points to illustrate this.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vector Addition
Vector addition is a fundamental concept in vector fields, allowing us to combine two or more vectors to create a new vector. In our exercise, we add vectors \( \vec{F} \) and \( b \vec{G} \) to produce another vector field \( \vec{L} = \vec{F} + b \vec{G} \).

This addition means combining the horizontal components \( (i) \) and the vertical components \( (j) \) of both vectors to find the i and j components of \( \vec{L} \).
  • If \( b = 0 \), the vector \( \vec{L} = \vec{F} \), hence it's entirely defined by \( \vec{F} \).
  • For \( b > 0 \), \( \vec{L} \) becomes \( (x - by) \vec{i} + (y + bx) \vec{j} \), where \( \vec{G} \) contributes a positive rotation component.
  • When \( b < 0 \), the contribution from \( \vec{G} \) is negative, modifying \( \vec{L} \) into \( (x - by) \vec{i} + (y + bx) \vec{j} \) but with a potential clockwise rotation effect due to the negative value.
Understanding vector addition helps illustrate how different factors (like \( b \) here) influence direction and magnitude.
Radial Vector Fields
A radial vector field is a special type of field where vectors radiate outward from or inward to a central point, often the origin. In our exercise, the case where \( b = 0 \) demonstrates a perfect example of a radial vector field.

The vector \( \vec{F} = x \vec{i} + y \vec{j} \) points straight from the origin towards any point \( (x, y) \). Each vector's length increases proportionally with distance from the origin, creating a field that looks like spokes on a wheel. Key characteristics of a radial vector field include:
  • Direction matching that of the position vector \((x, y)\).
  • Magnitude increasing as the distance from the origin increases.
  • Symmetry in all directions from the center.
In the absence of additional components (like \( b \vec{G} \)), this radial nature dominates, making it unmistakable in a vector field sketch.
Rotation in Vector Fields
Vector fields can include rotation, which refers to vectors having a directional tendency to rotate around a point. In the exercise, the vector \( \vec{G} = -y \vec{i} + x \vec{j} \) introduces a rotational aspect to our field due to its counter-clockwise orientation.

Rotation can be affected by various factors, such as:
  • The magnitude of the rotational vector components \((x, y)\).
  • The parameter \( b \), which scales the rotational component in \( \vec{L} = \vec{F} + b \vec{G} \).
For \( b > 0 \), this rotation occurs counter-clockwise because \( \vec{G} \)'s influence is positive. Conversely, with \( b < 0 \), the field rotates clockwise since the negative \( b \) subtracts from the counter-clockwise contribution of \( \vec{G} \). Visualizing these effects enhances your understanding of dynamic changes in vector fields.
Sketching Vector Fields
Sketching vector fields helps visualize how vectors behave across different regions of a plane. For this exercise, different values of \( b \) offer varying vector field dynamics, which we can represent through sketches.

To sketch each scenario, follow these steps:
  • For \( b = 0 \): Draw vectors radiating outwards at various points (like spokes from the origin). Each vector should be proportional to its position \((x, y)\).
  • When \( b > 0 \): Introduce a slight curve to the vectors following a counter-clockwise direction, looping around the origin.
  • For \( b < 0 \): Curve vectors in a clockwise direction, reflecting the negative rotation's impact.
The sketch's goal is to clearly show the trends and behaviors as \( b \) alters the vector field鈥檚 attitude. It's a valuable exercise for comprehending how vector fields operate in a visual and intuitive way.

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Most popular questions from this chapter

Are the statements true or false? Give reasons for your answer. The parametric curve \(x=t^{2}, y=t^{4}\) for \(0 \leq t \leq 1\) is a parabola.

Decide if the statement is true or false for all smooth parameterized curves \(\vec{r}(t)\) and all values of \(t\) for which \(\vec{r}^{\prime}(t) \neq \overrightarrow{0}\). The vector \(\vec{r}^{\prime}(t)\) is tangent to the curve at the point with position vector \(\vec{r}(t)\)

Are the statements true or false? Give reasons for your answer. Both \(x=-t+1, y=2 t\) and \(x=2 s, y=-4 s+2\) describe the same line.

A line has equation \(\vec{r}=\vec{a}+t \vec{b}\) where \(\vec{r}=x \vec{i}+y \vec{j}+z \vec{k}\) and \(\vec{a}\) and \(\vec{b}\) are constant vectors such that \(\vec{a} \neq \overrightarrow{0}, \vec{b} \neq\) \(\overrightarrow{0}, \vec{b}\) not parallel or perpendicular to \(\vec{a} .\) For each of the planes (a)-(c), pick the equation (i)-(ix) which represents it. Explain your choice. (a) A plane perpendicular to the line and through the origin. (b) A plane perpendicular to the line and not through the origin. (c) A plane containing the line. (i) \(\vec{a} \cdot \vec{r}=|| \vec{b}||\) (ii) \(\vec{b} \cdot \vec{r}=|| \vec{a}||\) (iii) \(\vec{a} \cdot \vec{r}=\vec{b} \cdot \vec{r}\) (iv) \((\vec{a} \times \vec{b}) \cdot(\vec{r}-\vec{a})=0\) (v) \(\vec{r}-\vec{a}=\vec{b}\) (vi) \(\vec{a} \cdot \vec{r}=0\) (vii) \(\vec{b} \cdot \vec{r}=0\) (viii) \(\vec{a}+\vec{r}=\vec{b}\) (ix) \((\vec{a} \times \vec{b}) \cdot(\vec{r}-\vec{b})=\|\vec{a}\|\)

You bicycle along a straight flat road with a safety light attached to one foot. Your bike moves at a speed of 25 \(\mathrm{km} / \mathrm{hr}\) and your foot moves in a circle of radius \(20 \mathrm{cm}\) centered \(30 \mathrm{cm}\) above the ground, making one revolution per second. (a) Find parametric equations for \(x\) and \(y\) which describe the path traced out by the light, where \(y\) is distance (in cm) above the ground. (b) Sketch the light's path. (c) How fast (in revolutions/sec) would your foot have to be rotating if an observer standing at the side of the road sees the light moving backward?
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