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Chapter 3: Problem 10

Find the (a) \(x\), (b) \(y\), and (c) z components of the sum \(\vec{r}\) of the displacements \(\vec{c}\) and \(\vec{d}\) whose components in meters are \(c_{x}=7.4, c_{y}=-3.8, c_{z}=-6.1 ; d_{x}=4.4, d_{y}=-2.0, d_{z}=3.3\).

Short Answer

Expert verified
\( r_x = 11.8 \), \( r_y = -5.8 \), \( r_z = -2.8 \)

Step by step solution

01

Understand the Given Vectors

We are given the components of two vectors \( \vec{c} \) and \( \vec{d} \). The components of \( \vec{c} \) are \( c_x = 7.4 \), \( c_y = -3.8 \), and \( c_z = -6.1 \). The components of \( \vec{d} \) are \( d_x = 4.4 \), \( d_y = -2.0 \), and \( d_z = 3.3 \). We need to find the vector \( \vec{r} = \vec{c} + \vec{d} \) and determine its individual components.
02

Calculate the x-component

To find the \( x \)-component of the resulting vector \( \vec{r} \), we need to add the \( x \)-components of \( \vec{c} \) and \( \vec{d} \). This is given by the equation \[ r_x = c_x + d_x = 7.4 + 4.4 \]. Calculating this gives \( r_x = 11.8 \).
03

Calculate the y-component

To find the \( y \)-component of the resulting vector \( \vec{r} \), we add the \( y \)-components of \( \vec{c} \) and \( \vec{d} \). This is given by the equation \[ r_y = c_y + d_y = -3.8 + (-2.0) \]. Calculating this results in \( r_y = -5.8 \).
04

Calculate the z-component

To find the \( z \)-component of the resulting vector \( \vec{r} \), we add the \( z \)-components of \( \vec{c} \) and \( \vec{d} \). This is given by the equation \[ r_z = c_z + d_z = -6.1 + 3.3 \]. Calculating this results in \( r_z = -2.8 \).
05

Summarize the Resulting Components

The resulting vector \( \vec{r} \) has the following components: - \( x \)-component: \( r_x = 11.8 \)- \( y \)-component: \( r_y = -5.8 \)- \( z \)-component: \( r_z = -2.8 \).

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Key Concepts

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

Understanding Components of Vectors
Vectors are quantities that have both magnitude and direction. They are fundamental in representing physical quantities like force, velocity, and displacement. Break a vector down into its components to understand it better.
  • A vector in three-dimensional space can be expressed in terms of its components along the x, y, and z axes.
  • Each component indicates how much influence the vector has in that particular direction.
For example, if we have a vector \( \vec{c} \) with components \( c_x = 7.4 \), \( c_y = -3.8 \), and \( c_z = -6.1 \), it tells us that \( \vec{c} \) has a positive influence along the x-axis but negative influences along the y and z axes.
Breaking vectors into components is especially helpful in solving physics problems as it allows us to handle each direction independently.
What is Displacement?
Displacement refers to the change in position of an object. It's a vector quantity, meaning it has both magnitude and direction.
  • Displacement is different from distance, which is scalar and only considers magnitude.
  • When multiple displacements are involved, each one can be treated as a vector with specific components.
In physics problems, displacement can usually be represented by vectors like \( \vec{c} \) and \( \vec{d} \), which denote the beginning and ending points of an object's journey. By analyzing the components of these vectors, you can determine the resultant path that the object takes in space.
Determining the Resultant Vector
A resultant vector is the vector that results from adding two or more vectors together. In our example, we need to find the resultant vector \( \vec{r} \) by summing vectors \( \vec{c} \) and \( \vec{d} \).To find the resultant vector:
  • Add their corresponding components: the x-components together, the y-components together, and the z-components together.
  • Use the formulas: \[ r_x = c_x + d_x \]
    \[ r_y = c_y + d_y \]
    \[ r_z = c_z + d_z \] These give us the components of \( \vec{r} \), which are \( r_x = 11.8 \), \( r_y = -5.8 \), and \( r_z = -2.8 \).
The resultant vector provides a single vector that accounts for the total displacement created by \( \vec{c} \) and \( \vec{d} \), simplifying further calculations.
Effective Physics Problem Solving Techniques
Physics problem-solving becomes easier when broken down into steps:
  • Understand the problem: Read the problem carefully and identify given data and what you need to find. In our case, this was recognizing that we had to find the components of the resultant vector \( \vec{r} \).
  • Break it down: Decompose the vector quantities into their components. This helps manage complex physical situations more easily.
  • Use appropriate equations: Apply vector addition rules to find unknown components. Each step in solving should logically follow from the last.
  • Review your work: Check calculations to ensure accuracy. In physics, small mistakes in computation can lead to significant errors in results.
By adhering to this systematic approach, physics problems become less intimidating, allowing you to achieve clarity and solutions more effectively.

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Most popular questions from this chapter

A displacement vector \(\vec{r}\) in the \(x y\) plane is \(12 \mathrm{~m}\) long and directed at angle \(\theta=30^{\circ}\) in Fig. 3-21. Determine (a) the \(x\) component and (b) the \(y\) component of the vector.

In a game of lawn chess, where pieces are moved between the centers of squares that are each \(1.00 \mathrm{~m}\) on edge, a knight is moved in the following way: (1) two squares forward, one square rightward; (2) two squares leftward, one square forward; (3) two squares forward, one square leftward. What are (a) the magnitude and (b) the angle (relative to "forward") of the knight's overall displacement for the series of three moves?

An explorer is caught in a whiteout (in which the snowfall is so thick that the ground cannot be distinguished from the sky) while returning to base camp. He was supposed to travel due north for \(4.8 \mathrm{~km}\), but when the snow clears, he discovers that he actually traveled \(7.8 \mathrm{~km}\) at \(50^{\circ}\) north of due east. (a) How far and (b) in what direction must he now travel to reach base camp?

If the \(x\) component of a vector \(\vec{a}\), in the \(x y\) plane, is half as large as the magnitude of the vector, find the tangent of the angle between the vector and the \(x\) axis.

(a) In unit-vector notation, what is the sum \(\vec{a}+\vec{b}\) if \(\vec{a}=(4.0 \mathrm{~m}) \hat{\mathrm{i}}+(3.0 \mathrm{~m}) \hat{\mathrm{j}}\) and \(\vec{b}=(-13.0 \mathrm{~m}) \hat{\mathrm{i}}+(7.0 \mathrm{~m}) \hat{\mathrm{j}} ?\) What are the (b) magnitude and (c) direction of \(\vec{a}+\vec{b}\) ?
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