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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 Magnitude of a Vector

The magnitude or norm of a vector \( \mathbf{v} \) is denoted as \( \| \mathbf{v} \| \). It is a measure of the vector's length. The operation \( \|k \mathbf{v}\| \) represents the magnitude of the vector after it has been scaled by the scalar \( k \).
02

Magnitude Being Zero

If the magnitude of a vector \( \|k \mathbf{v}\| \) is zero, then the vector itself must be the zero vector. This is because the magnitude of any non-zero vector is positive.
03

Analyzing \(k\) and \(\mathbf{v}\) Separately

The expression \( k \mathbf{v} = \mathbf{0} \) (where \( \mathbf{0} \) is the zero vector) implies either \( k = 0 \) or \( \mathbf{v} = \mathbf{0} \), or both. This is because the only way the scalar multiplication of a vector results in the zero vector is if the scalar is zero or the vector is zero.
04

Conclusion

From \( \|k \mathbf{v}\| = 0 \), we know that either \( k \) must be zero, or \( \mathbf{v} \) must be the zero vector, or possibly both conditions hold true.

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Key Concepts

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

Scalar Multiplication
In vector mathematics, scalar multiplication involves multiplying a vector by a scalar. A scalar is simply a real number, while a vector is an object that has both magnitude and direction. When we apply scalar multiplication, each component of the vector is multiplied by the scalar.
For example, if you have a vector \( \mathbf{v} = [x, y] \) and a scalar \( k \), then the resulting vector through scalar multiplication is \( k \mathbf{v} = [kx, ky] \). This operation scales the vector.
  • If \( k > 1 \), the vector becomes longer, stretching in the same direction as the original.
  • If \( 0 < k < 1 \), the vector shrinks but remains in the same direction.
  • If \( k = 0 \), no matter the vector \( \mathbf{v} \), the result is always the zero vector.
  • If \( k < 0 \), the vector not only changes length but also points in the opposite direction.
Understanding scalar multiplication is crucial for grasping more complex vector operations, such as when and why a vector's magnitude can become zero.
Zero Vector
The zero vector is a unique vector in mathematics. It is defined as having all its components equal to zero, symbolized as \( \mathbf{0} = [0, 0, \ldots, 0] \). The zero vector holds an important place in vector theory because it has a magnitude of zero.
No matter the number of dimensions, the zero vector acts as the additive identity in vector spaces. For any vector \( \mathbf{v} \), when you add the zero vector to it, you get the original vector back: \( \mathbf{v} + \mathbf{0} = \mathbf{v} \).
  • Geometrically, the zero vector denotes a point at the origin of the vector space.
  • During scalar multiplication, any non-zero vector multiplied by zero results in the zero vector.
  • The zero vector has a direction that is undefined because it lacks magnitude.
Recognizing the zero vector is helpful when determining conditions that lead to expressions like \( \|k \mathbf{v}\| = 0 \).
Norm of a Vector
The norm of a vector, often called its magnitude, is a measure of the vector's size or length. For a vector \( \mathbf{v} = [x_1, x_2, \ldots, x_n] \), its norm is calculated using the formula:\[ \| \mathbf{v} \| = \sqrt{x_1^2 + x_2^2 + \ldots + x_n^2} \]The norm is always a non-negative value.
Key points about vector norms include:
  • The norm is zero if and only if the vector itself is the zero vector.
  • Norms obey the triangle inequality, meaning that for any vectors \( \mathbf{u} \) and \( \mathbf{v} \), the inequality \( \| \mathbf{u} + \mathbf{v} \| \leq \| \mathbf{u} \| + \| \mathbf{v} \| \) holds.
  • Norms are invariant under scalar multiplication, adhering to the rule \( \| k \mathbf{v} \| = |k| \cdot \| \mathbf{v} \| \).
Understanding vector norms is critical for analyzing vector magnitudes and discovering why conditions such as \( \|k \mathbf{v}\| = 0 \) arise.

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Most popular questions from this chapter

Find an equation of the sphere with center \((2,1,-3)\) that is tangent to the plane \(x-3 y+2 z=4\)

Do the points \((1,0,-1),(0,2,3),(-2,1,1),\) and \((4,2,3)\) lie in the same plane? Justify your answer two different ways.

Let \(L\) be the line \(x=3 t+1, y=-5 t, z=t\) (a) Show that \(L\) lies in the plane \(2 x+y-z=2\) (b) Show that \(L\) is parallel to the plane \(x+y+2 z=0 .\) Is the line above, below, or on this plane?

Find parametric equations of the line of intersection of the planes. $$ \begin{array}{l}{3 x-5 y+2 z=0} \\ {z=0}\end{array} $$

Find an equation of the plane that satisfies the stated conditions. The plane through \((-1,4,-3)\) that is perpendicular to the line \(x-2=t, y+3=2 t, z=-t\)
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