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Magazine Chapter 11: Problem 38
What do you know about \(k\) and \(\mathbf{v}\) if \(\|k \mathbf{v}\|=0\) ?
Short Answer
Expert verified
Either \( k = 0 \), \( \mathbf{v} = \mathbf{0} \), or both.
Step by step solution
01
Understanding the Vector Norm
The expression \( \|k \mathbf{v}\| \) represents the norm (or length) of the vector \( k \mathbf{v} \). The norm of a vector \( \mathbf{v} \) is given by \( \| \mathbf{v} \| = \sqrt{v_1^2 + v_2^2 + abla v_n^2} \), where \( v_1, v_2, \dots, v_n \) are the components of the vector. For any non-zero vector, its norm is positive.
02
Analyzing the Given Norm Equation
The given condition \( \|k \mathbf{v}\|=0 \) means the norm of the vector \( k \mathbf{v} \) is zero. A vector has zero norm if and only if the vector itself is the zero vector.
03
Investigating the Zero Vector Condition
The vector \( k \mathbf{v} \) is the zero vector if each of its components is zero. This can happen if either \( k = 0 \) or \( \mathbf{v} = \mathbf{0} \) (or both). This is because scaling a vector by zero results in the zero vector regardless of the vector \( \mathbf{v} \), and any vector \( \mathbf{0} \) scaled by any scalar is still the zero vector.
04
Conclusion
From the analysis, for the norm \( \| k \mathbf{v} \| \) to equal zero, either the scalar \( k = 0 \) or the vector \( \mathbf{v} = \mathbf{0} \), or both, must be true. Therefore, these are the two possible conditions that satisfy the equation \( \|k \mathbf{v}\| = 0 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Zero Vector
A zero vector is a unique vector that has all of its components equal to zero. This can be visualized as a point at the origin in a coordinate system. In mathematical terms, for a vector \( \mathbf{v} \) with components \( (v_1, v_2, \ldots, v_n) \), the zero vector is represented as \( \mathbf{0} = (0, 0, \ldots, 0) \). A zero vector has a few distinct characteristics:
- Its norm (length) is zero, i.e., \( \| \mathbf{0} \| = 0 \).
- Adding a zero vector to any vector \( \mathbf{u} \) does not change the vector: \( \mathbf{u} + \mathbf{0} = \mathbf{u} \).
- It plays the same role in vector addition as the number 0 plays in arithmetic addition.
Scalars
In the context of vectors, scalars are simply regular numbers used to scale or multiply vectors. When you multiply a vector \( \mathbf{v} \) by a scalar \( k \), you change the vector's magnitude but not its direction (unless \( k \) is negative, which would reverse it). Here are some essential aspects of scalars:
- When \( k = 1 \), the vector remains unchanged: \( k \cdot \mathbf{v} = \mathbf{v} \).
- If \( k = 0 \), the resulting vector is the zero vector: \( k \cdot \mathbf{v} = \mathbf{0} \) for any vector \( \mathbf{v} \).
- A positive scalar multiplies the magnitude of the vector, while a negative scalar also reverses its direction.
Length of a Vector
The length of a vector, often called the norm, is a measure of how long the vector is. It uses the same principles as calculating the distance in geometry. For a vector \( \mathbf{v} = (v_1, v_2, \ldots, v_n) \), its length is represented by \( \| \mathbf{v} \| \) and calculated as:\[\| \mathbf{v} \| = \sqrt{v_1^2 + v_2^2 + \cdots + v_n^2}\]Some critical features of vector length include:
- A vector's length is always non-negative; it is zero if and only if the vector is the zero vector.
- It is invariant under a change of coordinate system, meaning the length does not depend on the perspective.
- The length gives a way to compare vectors. A larger norm indicates a longer vector.
