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Chapter 3: Problem 17

How would you add two vectors that are not perpendicular or parallel?

Short Answer

Expert verified
The resultant vector is founded by adding the x and y components of the vector independently, then calculating the magnitude and direction of the resultant from these totals.

Step by step solution

01

Decompose the Vectors

Break down each vector into its x and y components. This can be done using trigonometry since the directions of the vectors are known. For vector A with magnitude \( |A| \) and angle \( \theta_A \) relative to the x-axis, the x-component can be calculated as \( A_x = |A| \cos(\theta_A) \) and the y-component as \( A_y = |A| \sin(\theta_A) \). Repeat the same process for vector B with magnitude \( |B| \) and angle \( \theta_B \).
02

Sum the Components

Now sum the x-components of both vectors, and do the same for the y-components. That is, \( R_x = A_x + B_x \) and \( R_y = A_y + B_y \). These new components, \( R_x \) and \( R_y \), represent the x and y components of the resultant vector \( R \).
03

Calculate The Magnitude and Direction

Finally, use these components to calculate the magnitude and direction of the resultant vector. The magnitude can be found using Pythagorean theorem, \( |R| = \sqrt{ R_x^{2} + R_y^{2}} \), while the direction, \( \theta_R \), can be determined with the tangent function: \( \theta_R = \arctan( \frac{R_y}{R_x}) \). Adjust this angle as necessary according to quadrant rules.

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Key Concepts

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

Understanding Trigonometry in Vector Addition
Trigonometry is essential when adding vectors that are not at right angles. By using trigonometric functions, we can decompose vectors into their horizontal and vertical components.

For a vector with a known magnitude and angle relative to the x-axis, we use the cosine function to find the x-component, and the sine function to find the y-component. This is expressed as:
  • For the x-component: \( A_x = |A| \cos(\theta_A) \)
  • For the y-component: \( A_y = |A| \sin(\theta_A) \)
This process of breaking down vectors is crucial for analyzing their effect in different directions. Understanding these components is the first step in finding how vectors combine.
Finding the Resultant Vector
After decomposing each vector into its components, the next phase is to find the resultant vector, or the vector that represents the combined effect of the original vectors.

This is done by:
  • Adding up all the x-components: \( R_x = A_x + B_x \)
  • Adding up all the y-components: \( R_y = A_y + B_y \)
The vector represented by these summed components shows how the two original vectors interact in terms of direction and magnitude.
Angle Calculation With Vectors
Once the resultant vector's components are found, calculating its angle gives us a full understanding of its orientation.

The direction of this resultant vector relative to the x-axis is determined through the tangent function:
  • The angle \( \theta_R \) can be calculated as \( \theta_R = \arctan(\frac{R_y}{R_x}) \)
Remember to consider the signs of \( R_x \) and \( R_y \) to determine the correct quadrant for \( \theta_R \).

Quadrant checks are vital, as they adjust the angle to accurately reflect the vector's direction.
Applying the Pythagorean Theorem
The Pythagorean theorem is key to determining the magnitude of the resultant vector. By treating the x and y components as perpendicular sides of a right triangle, we can find the length of the hypotenuse, which is the vector's magnitude.

The formula to find this magnitude is: \[ |R| = \sqrt{R_x^2 + R_y^2} \]

This computation gives a clear measure of how strong or long the resultant vector is, based on the contributions of both the x and y components.

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Most popular questions from this chapter

If a person can jump a horizontal distance of \(3.0 \mathrm{m}\) on Earth, how far could the person jump on the moon, where the free-fall acceleration is \(g / 6\) and \(g=9.81 \mathrm{m} / \mathrm{s}^{2} ?\) How far could the person jump on Mars, where the acceleration due to gravity is \(0.38 g ?\)

The vector sum of three vectors gives a resultant equal to zero. What can you say about the vectors?

A place kicker must kick a football from a point \(36.0 \mathrm{m} \text { (about } 40.0 \mathrm{yd})\) from the goal. As a result of the kick, the ball must clear the crossbar, which is \(3.05 \mathrm{m}\) high. When kicked, the ball leaves the ground with a speed of \(20.0 \mathrm{m} / \mathrm{s}\) at an angle of \(53^{\circ}\) to the horizontal. a. By how much does the ball clear or fall short of clearing the crossbar? b. Does the ball approach the crossbar while still rising or while falling?

What is another way of saying \(-30 \mathrm{m} / \mathrm{s}\) west?

A student accurately uses the method for combining vectors. The two vectors she combines have magnitudes of 55 and 25 units. The answer that she gets is either \(85,20,\) or \(55 .\) Pick the correct answer, and explain why it is the only one of the three that can be correct.
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