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Doubling the Magnitude of a Vector and Its Components
Doubling the Magnitude of a Vector and Its Components
Understanding the behavior of vectors when their magnitudes change, especially while keeping their direction the same, is a fundamental concept in linear algebra and physics. This article explores the relationship between a vector's magnitude and the components that make it up, emphasizing the impact when the magnitude of a vector is doubled.
Introduction to Vectors and Magnitude
Vectors are mathematical objects used to represent quantities that have both magnitude and direction. In a coordinate system, a vector can be expressed in terms of its components along various axes. For instance, in a 3-dimensional Cartesian coordinate system, a vector (mathbf{v}) can be written as:
(mathbf{v} t_1 mathbf{b}_1 t_2 mathbf{b}_2 t_n mathbf{b}_n)where ({mathbf{b}_1, mathbf{b}_2, dots, mathbf{b}_n}) is an orthonormal basis. The magnitude (or length) of the vector (mathbf{v}), denoted as (lVert mathbf{v} rVert), is calculated as:
[lVert mathbf{v} rVert sqrt{t_1^2 t_2^2 dots t_n^2}]Effect of Doubling the Magnitude of a Vector
Suppose we have a vector (mathbf{v}) with components ({t_1, t_2, dots, t_n}). If the magnitude of (mathbf{v}) is doubled, how do the components of (mathbf{v}) change? To answer this, we start by noting that a vector (mathbf{w}) has the same direction as (mathbf{v}) if and only if there exists a real number (k geq 0) such that:
(mathbf{w} kmathbf{v} k(t_1 mathbf{b}_1 t_2 mathbf{b}_2 dots t_n mathbf{b}_n))
Given the orthonormality of the basis, the magnitude of (mathbf{w}) is:
[lVert mathbf{w} rVert^2 k^2 (t_1^2 t_2^2 dots t_n^2) k^2 lVert mathbf{v} rVert^2]Suppose the magnitude of (mathbf{v}) is doubled to become (lVert mathbf{w} rVert), then:
[lVert mathbf{w} rVert^2 4 lVert mathbf{v} rVert^2]This implies:
[k^2 4 implies k pm 2]For (mathbf{w}) to have the same direction as (mathbf{v}), (k) must be (2). Therefore, the components of (mathbf{v}) are doubled when its magnitude is doubled.
Examples and Further Clarifications
For a more concrete understanding, consider a 3-dimensional vector (mathbf{v} A_x hat{a}_x A_y hat{a}_y A_z hat{a}_z). If the magnitude of (mathbf{v}) is doubled while keeping its direction unchanged, the new vector (mathbf{w}) is:
(2 mathbf{v} 2(A_x hat{a}_x A_y hat{a}_y A_z hat{a}_z) 2A_x hat{a}_x 2A_y hat{a}_y 2A_z hat{a}_z)
This clearly shows that each component of (mathbf{v}) is doubled.
Conclusion
In summary, when the magnitude of a vector is doubled while its direction remains unchanged, every component of the vector also doubles. This relationship is fundamental in various fields, including physics, engineering, and computer science, where vectors are used to represent and analyze physical quantities.
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