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Chapter 2: Problem 20

If two vectors have the same magnitude, do their components have to be the same?

Short Answer

Expert verified
No, two vectors can have the same magnitude without having the same components.

Step by step solution

01

Understanding vectors

A vector is a quantity that has both magnitude and direction. In a coordinate system, each vector is characterized by its components. For a 2-dimensional vector, these are often referred to as the x-component and the y-component.
02

The relation between vector components and magnitude

The magnitude of a 2-dimensional vector with components (x, y) is given by the Pythagorean theorem: \( \sqrt{x^2 + y^2} \). Two vectors can have the same magnitude but different x and y components. For example, (3, 4) and (4, 3) are two vectors with the same magnitude but different components.
03

Conclusion

Since vectors can have the same magnitude but different components, it is therefore possible for two vectors to have the same magnitude without having the same components.

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Key Concepts

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

Vector Components
Every vector can be broken down into its fundamental parts. These parts are called its components. To visualize this, think of each component as an arrow pointing in the direction of each axis in a coordinate system. The combined effect of these components gives the original vector.
  • For a 2-dimensional space, there are typically two vector components: the x-component and the y-component.
  • The components represent the vector's influence along the horizontal and vertical directions in a graph.
  • Each component is essentially the projection of the vector along its respective axis.
Understanding vector components is crucial because they help describe how a vector behaves in a coordinate system. They also allow for easier calculations such as finding a vector's magnitude or performing vector addition.
Pythagorean Theorem
The Pythagorean theorem is a mathematical rule that relates the lengths of the sides of a right triangle. For vectors, this theorem is a key tool to calculate the magnitude from its components.
  • To find the magnitude of a vector, consider the components as the two shorter sides of a right triangle.
  • The formula derived from the Pythagorean theorem to find the magnitude (hypotenuse) is: \( \sqrt{x^2 + y^2} \).
  • This tells us how long the vector is in terms of distance, irrespective of its direction.
Even if two vectors have different components, they can have the same magnitude if the squared sum of their respective components is identical. For instance, vectors (3, 4) and (4, 3) both result in a magnitude of 5.
Coordinate System
A coordinate system is a method of specifying points using numbers in a space. When dealing with vectors, it's important to understand the role of a coordinate system.
  • In a typical 2D coordinate system, we have an x-axis (horizontal) and a y-axis (vertical).
  • A vector in this system is often described by how it extends along each of these axes, which are its components.
  • By plotting a vector's components on the coordinate system, we can easily visualize its direction and magnitude.
Coordinate systems allow us to translate real-world problems into mathematical models. These models can then be solved or analyzed using vector operations. Understanding how a vector fits into its coordinate system helps us foresee how it interacts with other vectors and objects in that space.

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Most popular questions from this chapter

A small plane flies \(40.0 \mathrm{km}\) in a direction \(60^{\circ}\) north of east and then flies \(30.0 \mathrm{km}\) in a direction \(15^{\circ}\) north of east. Use the analytical method to find the total distance the plane covers from the starting point, and the geographic direction of its displacement vector. What is its displacement vector?

What is the dot product of a vector with the cross product that this vector has with another vector?

Does the odometer in an automobile indicate a scalar or a vector quantity?

If one of the two components of a vector is not zero, can the magnitude of the other vector component of this vector be zero?

A particle undergoes three consecutive displacements given by vectors $$\overrightarrow{\mathbf{D}}_{1}=(3.0 \hat{\mathbf{i}}-4.0 \hat{\mathbf{j}}-2.0 \hat{\mathbf{k}}) \mathrm{mm}$$ $$\overrightarrow{\mathbf{D}}_{2}=(1.0 \hat{\mathbf{i}}-7.0 \hat{\mathbf{j}}+4.0 \hat{\mathbf{k}}) \mathrm{mm}$$ , and $$\overrightarrow{\mathbf{D}}_{3}=(-7.0 \hat{\mathbf{i}}+4.0 \hat{\mathbf{j}}+1.0 \hat{\mathbf{k}}) \mathrm{mm}$$ (a) Find the resultant displacement vector of the particle. (b) What is the magnitude of the resultant displacement? (c) If all displacements were along one line, how far would the particle travel?
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