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Chapter 1: Problem 36

The displacement vector \(\overrightarrow{\mathrm{A}}\) has scalar components of \(A_{x}=80.0 \mathrm{~m}\) and \(A_{y}=60.0 \mathrm{~m}\) The displacement vector \(\overrightarrow{\mathbf{B}}\) has a scalar component of \(B_{x}=60.0 \mathrm{~m}\) and a magnitude of \(B=75.0 \mathrm{~m} .\) The displacement vector \(\overrightarrow{\mathrm{C}}\) has a magnitude of \(C=100.0 \mathrm{~m}\) and is directed at an angle of \(36.9^{\circ}\) above the \(+x\) axis. Two of these vectors are equal. Determine which two, and support your choice with a calculation.

Short Answer

Expert verified
Vectors \( \overrightarrow{\mathrm{A}} \) and \( \overrightarrow{\mathrm{C}} \) are equal.

Step by step solution

01

Analyze Vector A

Vector \( \overrightarrow{\mathrm{A}} \) has components \( A_{x} = 80.0 \text{ m} \) and \( A_{y} = 60.0 \text{ m} \). We can calculate its magnitude using the formula \( A = \sqrt{A_{x}^2 + A_{y}^2} \). Substituting the given values gives us: \( A = \sqrt{80.0^2 + 60.0^2} = \sqrt{6400 + 3600} = \sqrt{10000} = 100.0 \text{ m} \). Thus, the magnitude of vector \( \overrightarrow{\mathrm{A}} \) is 100.0 m.
02

Analyze Vector B

Vector \( \overrightarrow{\mathbf{B}} \) has a component \( B_{x} = 60.0 \text{ m} \) and a given magnitude \( B = 75.0 \text{ m} \). To find the \( B_{y} \) component, we use the Pythagorean theorem: \( B = \sqrt{B_{x}^2 + B_{y}^2} \). Solving for \( B_{y} \) gives us: \( B_{y} = \sqrt{B^2 - B_{x}^2} = \sqrt{75.0^2 - 60.0^2} = \sqrt{5625 - 3600} = \sqrt{2025} = 45.0 \text{ m} \). So, \( B_{y} = 45.0 \text{ m} \), and the magnitude confirms as 75.0 m.
03

Calculate Components of Vector C

Vector \( \overrightarrow{\mathrm{C}} \) has a magnitude of \( C = 100.0 \text{ m} \) and an angle of \( 36.9^{\circ} \) from the \( +x \) axis. Calculate its components using trigonometry: \( C_{x} = C \cos(36.9^{\circ}) = 100.0 \cos(36.9^{\circ}) \approx 80.0 \text{ m} \) and \( C_{y} = C \sin(36.9^{\circ}) = 100.0 \sin(36.9^{\circ}) \approx 60.0 \text{ m} \). Thus, \( C_{x} \approx 80.0 \text{ m} \) and \( C_{y} \approx 60.0 \text{ m}. \)
04

Identify Equal Vectors

Vector \( \overrightarrow{\mathrm{A}} \) has components \( A_{x} = 80.0 \text{ m}, \ A_{y} = 60.0 \text{ m} \) and magnitude 100.0 m. Vector \( \overrightarrow{\mathrm{C}} \), calculated in Step 3, also has \( C_{x} \approx 80.0 \text{ m}, \ C_{y} \approx 60.0 \text{ m} \) and magnitude 100.0 m. Both have identical components and magnitude, thus \( \overrightarrow{\mathrm{A}} = \overrightarrow{\mathrm{C}} \).

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Key Concepts

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

Displacement Vectors
Displacement vectors are fundamental in physics and mathematics. They represent the change in position of an object. Each displacement vector has both magnitude and direction.
For instance, if you travel 3 blocks north and 4 blocks east, your total journey can be described as a single displacement vector, regardless of the path taken.
  • The direction is crucial, as it indicates the overall angle of the journey.
  • The magnitude tells just how far the starting and ending points are from each other without regard to the path.
This concept can simplify complex paths into simpler calculations, as seen with displacement vectors \( \overrightarrow{\mathrm{A}} \) and \( \overrightarrow{\mathrm{C}} \) from the exercise, which resulted in equal vectors.
Displacement vectors thus serve as a bridge between physical movement and mathematical representation.
Vector Magnitude Calculation
To fully understand a vector, we need to calculate its magnitude. This is the length of the displacement vector from its origin to its endpoint.
The formula for calculating the magnitude when you have the components \( A_x \) and \( A_y \) is straightforward:\[A = \sqrt{A_x^2 + A_y^2}\]
In our example, vector \( \overrightarrow{\mathrm{A}} \) has components 80.0 m and 60.0 m, leading to a magnitude of 100.0 m, confirmed through calculation.
  • This indicates a tight relationship between its components and true distance moved.
  • You can apply similar calculations to any vector with known components.
Thus, magnitude works as a scalar quantity, representing the size of a vector, independent from its direction.
Trigonometric Vector Components
Understanding vector components with trigonometry allows for breaking down a vector into its horizontal (\(x\)) and vertical (\(y\)) parts.
The process involves using sine and cosine trigonometric functions based on the vector鈥檚 angle with the \(x\)-axis:
- For the \(x\)-component: \[C_x = C \cos(\theta)\]- For the \(y\)-component: \[C_y = C \sin(\theta)\]
In the exercise, we saw vector \( \overrightarrow{\mathrm{C}} \) expressed through trigonometric calculations, yielding components of approximately 80.0 m and 60.0 m.
These methods highlight that:
  • Angles directly influence the size of \(x\) and \(y\) components.
  • The sine function aligns with the vertical distance, while cosine aligns with the horizontal.
This intuitive breakdown based on trigonometry is crucial in resolving the full impact of a vector in any given direction.

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Most popular questions from this chapter

You are driving into St. Louis, Missouri, and in the distance you see the famous Gateway-to-the-West arch. This monument rises to a height of \(192 \mathrm{~m}\). You estimate your line of sight with the top of the arch to be \(2.0^{\circ}\) above the horizontal. Approximately how far (in kilometers) are you from the base of the arch?

A regular tetrahedron is a three-dimensional object that has four faces, each of which is an equilateral triangle. Each of the edges of such an object has a length \(L\). The height \(H\) of a regular tetrahedron is the perpendicular distance from one corner to the center of the opposite triangular face. Show that the ratio between \(H\) and \(L\) is \(H / L=\sqrt{2 / 3}\).

(a) Considering the fact that \(3.28 \mathrm{ft}=1 \mathrm{~m}\), which is the larger unit for measuring area, \(1 \mathrm{ft}^{2}\) or \(1 \mathrm{~m}^{2} ?\) (b) Consider a \(1330-\mathrm{ft}^{2}\) apartment. With your answer to part (a) in mind and without doing any calculations, decide whether this apartment has an area that is greater than or less than \(1330 \mathrm{~m}^{2}\). In a \(1330-\mathrm{ft}^{2}\) apartment, how many square meters of area are there? Be sure that your answer is consistent with your answers to the Concept Questions.

A person is standing at the edge of the water and looking out at the ocean (see the drawing). The height of the person's eyes above the water is \(h=1.6 \mathrm{~m},\) and the radius of the earth is \(R=6.38 \times 10^{6} \mathrm{~m}\). (a) How far is it to the horizon? In other words, what is the distance \(d\) from the person's eyes to the horizon? (Note: At the horizon the angle between the line of sight and the radius of the earth is \(\left.90^{\circ} .\right)\) (b) Express this distance in miles.

Vector \(\overrightarrow{\text { A }}\) points along the \(+y\) axis and has a magnitude of 100.0 units. Vector \(\overrightarrow{\mathbf{B}}\) points at an angle of \(60.0^{\circ}\) above the \(+x\) axis and has a magnitude of 200.0 units. Vector \(\overrightarrow{\mathrm{C}}\) points along the \(+x\) axis and has a magnitude of 150.0 units. Which vector has (a) the largest \(x\) component and (b) the largest \(y\) component?
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