Chapter 4: Problem 53
Can a component of a vector be greater than the vector's magnitude?
Short Answer
Expert verified
No, a component of a vector cannot be greater than the vector's magnitude.
Step by step solution
01
Understanding Vector Magnitude and Components
A vector has both magnitude and direction. The magnitude of a vector is a measure of its length, which can be computed using the Pythagorean theorem for its components. If a vector is represented as \( \vec{v} = (v_x, v_y) \), its magnitude \( \|\vec{v}\| \) is given by the formula \( \|\vec{v}\| = \sqrt{v_x^2 + v_y^2} \).
02
Analyzing Components vs. Magnitude
The components of a vector, \( v_x \) and \( v_y \), are the projections of the vector along the coordinate axes. These components are individually \( v_x = \|\vec{v}\| \cos \theta \) and \( v_y = \|\vec{v}\| \sin \theta \), where \( \theta \) is the angle the vector makes with the horizontal axis.
03
Applying Inequalities to Magnitude and Components
Since \( \cos \theta \) and \( \sin \theta \) are both always between \(-1\) and \(1\), each component of the vector is less than or equal to the magnitude of the vector. Therefore, neither \( v_x \) nor \( v_y \) can exceed \( \|\vec{v}\| \).
04
Conclusion Based on Analysis
From the relationship that each component is a result of multiplying the magnitude by a cosine or sine which is bound between \(-1\) and \(1\), it follows that the component of a vector cannot be greater than the magnitude of the vector.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Vector Components
Vector components are crucial for understanding how vectors behave in a coordinate system. A vector in a plane can be expressed using its components along the coordinate axes. Let's break it down: imagine a vector as an arrow; this arrow can be represented in terms of how much it stretches along the horizontal axis ( \(v_x\)) and the vertical axis ( \(v_y\)).
Each component is a projection of the vector onto the respective axis:
Each component is a projection of the vector onto the respective axis:
- The horizontal component ( \(v_x\)) shows how far the vector moves along the x-axis.
- The vertical component ( \(v_y\)) reflects the vector's movement along the y-axis.
Pythagorean Theorem
The Pythagorean theorem is a foundational principle in geometry, particularly when calculating the magnitude of vectors. In the context of vectors, if you have a right triangle formed by the vector and its components, the Pythagorean theorem allows for calculating the vector's magnitude.
For a vector \(\vec{v} = (v_x, v_y)\), the magnitude \(\|\vec{v}\|\) is computed as:\[\|\vec{v}\| = \sqrt{v_x^2 + v_y^2}\]This equation shows that the magnitude or length of the vector is the hypotenuse of a right triangle with sides \(v_x\) and \(v_y\). This relationship makes it clear that the magnitude is always the longest side, meaning neither component can be greater than the magnitude itself.
For a vector \(\vec{v} = (v_x, v_y)\), the magnitude \(\|\vec{v}\|\) is computed as:\[\|\vec{v}\| = \sqrt{v_x^2 + v_y^2}\]This equation shows that the magnitude or length of the vector is the hypotenuse of a right triangle with sides \(v_x\) and \(v_y\). This relationship makes it clear that the magnitude is always the longest side, meaning neither component can be greater than the magnitude itself.
Trigonometric Functions
Trigonometric functions, specifically sine and cosine, play a key role in determining vector components. When a vector is expressed in terms of its direction and magnitude, the angle \(\theta\) it makes with the horizontal axis helps in calculating its components.
- The cosine of the angle (\(\cos \theta\)) helps determine the horizontal component \(v_x = \|\vec{v}\| \cos \theta\).
- The sine of the angle (\(\sin \theta\)) is used to find the vertical component \(v_y = \|\vec{v}\| \sin \theta\).
Vector Analysis
Vector analysis involves dissecting a vector into its components and comprehending the relationships among these components, their magnitude, and direction. This analysis informs whether a vector component can ever be greater than the vector's magnitude itself.
- Each component's value derives from a combination of both the vector's magnitude and a trigonometric function.
- The mathematical properties of sine and cosine mean that when multiplied by any magnitude, the resulting component is inherently bound to be less than or equal to the magnitude.