/*! 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 21 Three vectors \(\vec{a}\), and \... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 21

Three vectors \(\vec{a}\), and \(\vec{b}\), and \(\vec{c}\) each have a magnitude of \(50 \mathrm{~m}\) and lie in an \(x y\) plane. Their directions relative to the positive direction of the \(x\) axis are \(30^{\circ}, 195^{\circ}\), and \(315^{\circ}\), respectively. What are (a) the magnitude and (b) the angle of the vector \(\vec{a}+\vec{b}+\) \(\vec{c}\), and \((\mathrm{c})\) the magnitude and \((\mathrm{d})\) the angle of \(\vec{a}-\vec{b}+\vec{c} ?\) What are (e) the magnitude and (f) the angle of a fourth vector \(\vec{d}\) such that \((\vec{a}+\vec{b})-(\vec{c}+\vec{d})=0\)

Short Answer

Expert verified
(a) Use components R_x and R_y from Step 2 for the magnitude and Step 4 for the angle. (b) Use components S_x and S_y from Step 5 for the magnitude and Step 7 for the angle. (c) Use components d_x and d_y from Step 8 for the magnitude and Step 10 for the angle.

Step by step solution

01

- Calculate Components of Each Vector

Find the x and y components of the vectors \(\vec{a}\) \, \(\vec{b}\) \, and \(\vec{c}\).\ For \(\vec{a}\): \[a_x = 50 \cos(30^{\circ})\] and \[a_y = 50 \sin(30^{\circ}),\] For \(\vec{b}\): \[b_x = 50 \cos(195^{\circ})\] and \[b_y = 50 \sin(195^{\circ}),\] For \(\vec{c}\): \[c_x = 50 \cos(315^{\circ})\] and \[c_y = 50 \sin(315^{\circ}).\]
02

- Sum Components for \(\vec{a} + \vec{b} + \vec{c}\)

Calculate the x and y components of \(\vec{a} + \vec{b} + \vec{c}\).\ \[R_x = a_x + b_x + c_x\] and \[R_y = a_y + b_y + c_y.\]
03

- Find Magnitude of \(\vec{R} = \vec{a} + \vec{b} + \vec{c}\)

Use Pythagorean theorem to find the magnitude: \[|\vec{R}| = \sqrt{R_x^2 + R_y^2}.\]
04

- Find Angle of \(\vec{R} = \vec{a} + \vec{b} + \vec{c}\)

Calculate the angle using: \[\theta_R = \tan^{-1}\left(\frac{R_y}{R_x}\right).\]
05

- Sum Components for \(\vec{a} - \vec{b} + \vec{c}\)

Calculate the x and y components of \(\vec{a} - \vec{b} + \vec{c}\).\ \[S_x = a_x - b_x + c_x\] and \[S_y = a_y - b_y + c_y.\]
06

- Find Magnitude of \(\vec{S} = \vec{a} - \vec{b} + \vec{c}\)

Use Pythagorean theorem to find the magnitude: \[|\vec{S}| = \sqrt{S_x^2 + S_y^2}.\]
07

- Find Angle of \(\vec{S} = \vec{a} - \vec{b} + \vec{c}\)

Calculate the angle using: \[\theta_S = \tan^{-1}\left(\frac{S_y}{S_x}\right).\]
08

- Determine Components of \(\vec{d}\)

Solve \(\vec{a} + \vec{b} - \vec{c} - \vec{d} = 0\) for \(\vec{d}\).\ Thus, \[d_x = a_x + b_x - c_x\] and \[d_y = a_y + b_y - c_y.\]
09

- Find Magnitude of \(\vec{d}\)

Use Pythagorean theorem to find the magnitude: \[|\vec{d}| = \sqrt{d_x^2 + d_y^2}.\]
10

- Find Angle of \(\vec{d}\)

Calculate the angle using: \[\theta_d = \tan^{-1}\left(\frac{d_y}{d_x}\right).\]

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Key Concepts

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

Vector Components
To solve problems involving vectors, it's essential to break them down into components. Each vector has both an x-component and a y-component. For a vector \(\vec{a}\) whose magnitude is \(|a| = 50 \texttt{m}\) and direction is \(30^\texttt{掳}\) from the x-axis, we use trigonometry to find the components: \(a_x = 50 \texttt{cos}(30^\texttt{掳})\) and \(a_y = 50 \texttt{sin}(30^\texttt{掳})\). Breaking the vector down into these two parts helps in understanding its motion along the x and y axes separately. Similarly, we do this for other vectors like \(\vec{b}\) and \(\vec{c}\) to find their components based on their respective angles.
Magnitude Calculation
The magnitude of a vector reflects its length or size. When we sum or subtract vectors, we first find their resultant components. For example, to get the resultant vector \(\vec{R}\) from \(\vec{a} + \vec{b} + \vec{c}\), we sum up their x and y components separately: \(R_x = a_x + b_x + c_x\) and \(R_y = a_y + b_y + c_y\). After finding the resultant components, we apply the Pythagorean theorem to determine the magnitude: \[|\vec{R}| = \sqrt{R_x^2 + R_y^2}\]. This formula gives us the overall length of the resultant vector in the same way that you'd find the hypotenuse of a right triangle.
Angle Determination
After finding the magnitude of a vector, the next step is to determine its direction relative to the x-axis. This is done using the arctangent function. For the resultant vector \(\vec{R}\), if we know the components \(R_x\) and \(R_y\), the angle \(\theta_{\vec{R}}\) can be found using: \[\theta_{\vec{R}} = \texttt{tan}^{-1}(\frac{R_y}{R_x})\]. This angle tells us the vector's orientation in the plane. It's crucial to keep the signs of the components in mind, as they determine the correct quadrant for the angle.
Pythagorean Theorem
The Pythagorean theorem is a staple in vector operations. It's used to calculate the magnitude of a resultant vector. Given a right triangle with legs representing the vector's x and y components, the theorem states that \(c^2 = a^2 + b^2\), where \(c\) is the hypotenuse. Applied to vectors, if you have components \(R_x\) and \(R_y\), the magnitude \(R\) is \[|\vec{R}| = \sqrt{R_x^2 + R_y^2}\]. This principle forms the basis for determining the length of any vector in a two-dimensional space, providing a clear geometric interpretation of vector addition or subtraction.

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Most popular questions from this chapter

Two beetles run across flat sand, starting at the same point. Beetle 1 runs \(0.50 \mathrm{~m}\) due east, then \(0.80 \mathrm{~m}\) at \(30^{\circ}\) north of due east. Beetle 2 also makes two runs; the first is \(1.6 \mathrm{~m}\) at \(40^{\circ}\) east of due north. What must be (a) the magnitude and (b) the direction of its second run if it is to end up at the new location of beetle \(1 ?\)

Find the sum of the following four vectors in (a) unit-vector notation, and as (b) a magnitude and (c) an angle relative to \(+x\). \(\vec{P}: 10.0 \mathrm{~m}\), at \(25.0^{\circ}\) counterclockwise from \(+x\) \(\vec{Q}: 12.0 \mathrm{~m}\), at \(10.0^{\circ}\) counterclockwise from \(+y\) \(\vec{R}: 8.00 \mathrm{~m}\), at \(20.0^{\circ}\) clockwise from \(-y\) \(\vec{S}: 9.00 \mathrm{~m}\), at \(40.0^{\circ}\) counterclockwise from \(-y\)

A fire ant, searching for hot sauce in a picnic area, no goes through three displacements along level ground: \(\vec{d}_{1}\) for \(0.40 \mathrm{~m}\) southwest (that is, at \(45^{\circ}\) from directly south and from directly west), \(\vec{d}_{2}\) for \(0.50 \mathrm{~m}\) due east (that is, directly east), \(\vec{d}_{3}\) for \(0.60 \mathrm{~m}\) at \(60^{\circ}\) north of east (that is \(60.0^{\circ}\) toward the north from due east). Let the positive \(x\) direction be east and the positive \(y\) direction be north. What are (a) the \(x\) -component and (b) the \(y\) -component of \(\vec{d}_{1} ?\) What are (c) the \(x\) -component and (d) the \(y\) -component of \(\vec{d}_{2} ?\) What are (e) the \(x\) -component and (f) the \(y\) -component of \(\vec{d}_{3}\) ? What are \((\mathrm{g})\) the \(x\) -component, (h) the \(y\) -component, (i) the magnitude, and (j) the direction of the ant's net displacement? If the ant is to return directly to the starting point, \((\mathrm{k})\) how far and (I) in what direction should it move?

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

A protester carries his sign of protest \(40 \mathrm{~m}\) along a straight path, then \(20 \mathrm{~m}\) along a perpendicular path to his left, and then \(25 \mathrm{~m}\) up a water tower. (a) Choose and describe a coordinate system for this motion. In terms of that system and in unitvector notation, what is the displacement of the sign from start to end? (b) The sign then falls to the foot of the tower. What is the magnitude of the displacement of the sign from start to this new end?
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