/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 63 The components of a vector \(\ov... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 63

The components of a vector \(\overrightarrow{\mathbf{B}}\) satisfy \(B_{x}>0\) and \(B_{y}<0\). Is the direction angle of \(\overrightarrow{\mathbf{B}}\) between \(0^{\circ}\) and \(90^{\circ},\) between \(90^{\circ}\) and \(180^{\circ},\) between \(180^{\circ}\) and \(270^{\circ},\) or between \(270^{\circ}\) and \(360^{\circ} ?\)

Short Answer

Expert verified
The direction angle of \(\overrightarrow{\mathbf{B}}\) is between \(270^{\circ}\) and \(360^{\circ}\).

Step by step solution

01

Analyze the Quadrant

Vectors are represented in a coordinate plane divided into four quadrants. Since \(B_x > 0\), the vector has a positive x-component. This means the vector points to the right. Also, since \(B_y < 0\), the vector has a negative y-component, indicating it points downward. We conclude that the vector \(\overrightarrow{\mathbf{B}}\) lies in the Fourth Quadrant.
02

Determine the Range of Angle

The angles in a coordinate plane are measured starting from the positive x-axis in a counterclockwise direction. In the Fourth Quadrant, angles are measured from \(270^{\circ}\) to \(360^{\circ}\). Therefore, the direction angle of vector \(\overrightarrow{\mathbf{B}}\) is between \(270^{\circ}\) and \(360^{\circ}\).

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Key Concepts

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

Coordinate Plane
The coordinate plane is the foundation for understanding vector components and their behaviors. Imagine it like a graph paper where you can plot points, lines, and shapes based on their mathematical relationships. The plane is a two-dimensional surface formed by two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). Together, these axes allow us to locate points described by pairs of numbers known as coordinates. Each point on the coordinate plane is identified by an ordered pair
  • The first number in the pair, usually labeled as \(x\), indicates how far the point is from the y-axis.
  • The second number, \(y\), shows the distance from the x-axis.
The coordinate plane is essential for plotting vectors and analyzing their direction and magnitude. By understanding the coordinate plane, we can visually and mathematically represent the behavior and properties of vectors in the plane.
Quadrant System
When working with a coordinate plane, it is divided into four quadrants. This quadrant system helps us understand where certain points or vectors are located based on their coordinates. Below is how each quadrant is defined:
  • The First Quadrant: Where both \(x\) and \(y\) are positive.
  • The Second Quadrant: Where \(x\) is negative, and \(y\) is positive.
  • The Third Quadrant: Where both \(x\) and \(y\) are negative.
  • The Fourth Quadrant: Where \(x\) is positive, and \(y\) is negative.
Understanding this quadrant system is crucial in identifying the direction and location of vectors. For instance, a vector with a positive \(x\) component and a negative \(y\) component鈥攍ike in the given exercise鈥攆alls into the Fourth Quadrant. Recognizing the quadrant can instantly guide you to the possible range of direction angles that vector might have in the coordinate plane.
Direction Angle
The concept of direction angle is key when dealing with vectors in the coordinate plane. The direction angle of a vector is measured from the positive x-axis (to the right) counterclockwise. This is a conventional way to define the angle a vector makes with the horizontal axis.
  • In the First Quadrant, these angles range from 0掳 to 90掳.
  • In the Second Quadrant, angles span from 90掳 to 180掳.
  • In the Third Quadrant, these angles fall between 180掳 and 270掳.
  • In the Fourth Quadrant, direction angles are located between 270掳 and 360掳.
In the exercise, we found that the vector \(\overrightarrow{\mathbf{B}}\) is in the Fourth Quadrant, so its direction angle lies between 270掳 and 360掳. Knowing the quadrant gives a direct clue about the range of the direction angle, which simplifies the process of understanding vector directions.

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Most popular questions from this chapter

As an airplane taxies on the runway with a speed of \(16.5 \mathrm{m} / \mathrm{s}\), a flight attendant walks toward the tail of the plane with a speed of \(1.22 \mathrm{m} / \mathrm{s}\). What is the flight attendant's speed relative to the ground?

A football is thrown horizontally with an initial velocity of \((16.6 \mathrm{m} / \mathrm{s}) \hat{\mathrm{x}} .\) Ignoring air resistance, the average acceleration of the football over any period of time is \(\left(-9.81 \mathrm{m} / \mathrm{s}^{2}\right) \hat{y}\) (a) Find the velocity vector of the ball \(1.75 \mathrm{s}\) after it is thrown. (b) Find the magnitude and direction of the velocity at this time.

In its daily prowl of the neighborhood, a cat makes a displacement of \(120 \mathrm{m}\) due north, followed by a \(72-\mathrm{m}\) displacement due west. (a) Find the magnitude and direction of the displacement required for the cat to return home. (b) If, instead, the cat had first prowled \(72 \mathrm{m}\) west and then \(120 \mathrm{m}\) north, how would this affect the displacement needed to bring it home? Explain.

You throw a ball upward with an initial speed of \(4.5 \mathrm{m} / \mathrm{s}\). When it returns to your hand 0.92 s later, it has the same speed in the downward direction (assuming air resistance can be ignored). What was the average acceleration vector of the ball?

It is given that \(\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}=(-51.4 \mathrm{m}) \hat{\mathrm{x}}, \overrightarrow{\mathrm{C}}=(62.2 \mathrm{m}) \hat{\mathrm{x}},\) and \(\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}+\overrightarrow{\mathbf{C}}=(13.8 \mathrm{m}) \hat{\mathrm{x}} .\) Find the vectors \(\overrightarrow{\mathbf{A}}\) and \(\overline{\mathbf{B}}\)
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