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Chapter 11: Problem 106

State the definition of parallel vectors.

Short Answer

Expert verified
Parallel vectors are vectors that have the same or opposite direction. It can be mathematically stated as if a vector \(a\) is a scalar multiple of another vector \(b\), then they are parallel. The scalar \(\lambda\) can be non-zero, positive or negative.

Step by step solution

01

Definition

Two vectors are said to be parallel if they have the same or opposite direction. In other words, two vectors \(a\) and \(b\) are parallel if there exists a non-zero scalar \(\lambda\) such that \(a = \lambda b\). This is called the scalar multiple of vector \(b\). The parameter \(\lambda\) can be positive, which means that the vectors are parallel and oriented in the same direction, or negative, which means that they're parallel and oriented in opposite directions.
02

Additional Information

In plain language, if the directions of two vectors are the same or opposite, they are considered parallel. It's important to note that the magnitudes (or lengths) of the vectors do not have to be equal for the vectors to be parallel. They can be different lengths and still be parallel as long as they point in the exact same or directly opposite direction.

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Key Concepts

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

Scalar Multiple
A scalar multiple is a fundamental concept when discussing parallel vectors. Imagine you have a vector, let's call it \(\mathbf{b}\). When two vectors are parallel, it means one can be expressed as a multiple of the other. This is where the scalar, denoted as \(\lambda\), comes into play. Here, \(\lambda\) is a real number that scales the vector \(\mathbf{b}\) to transform it into another vector \(\mathbf{a}\).
This relationship is captured in the equation \(\mathbf{a} = \lambda \mathbf{b}\). If \(\lambda\) is positive, both vectors have the same direction. If it is negative, they point in opposite directions. To better grasp this, consider these essential points:
  • The scalar \(\lambda\) does not alter the direction of the vector, just its magnitude and orientation.
  • If \(\lambda = 0\), the result is a zero vector, meaning no actual direction or magnitude, which is why \(\lambda\) must be non-zero for parallel vectors.
  • A scalar multiple helps us identify and confirm the parallel nature between two vectors.
Vector Direction
A vector's direction is what defines its orientation in space. For vectors to be parallel, they need to either have the same direction or be direct opposites of each other. This simply means they point straight along the same line.
Think of it like two arrows flying parallel to each other, either going in the exact same direction or directly opposing each other without diverging from the path.
Key aspects of vector direction include:
  • Determining direction involves understanding the angle they make with a reference axis. For parallel vectors, their angle is equivalent or supplementary.
  • Parallel vectors maintain a constant angle between each other, ensuring consistent parallelism regardless of their magnitude differences.
  • It's crucial to realize that while vectors may vary in length, as long as their direction aligns, they are parallel.
Non-Zero Scalar
A non-zero scalar is a vital requirement in defining parallel vectors. The definition itself states that there exists a scalar \(\lambda\) which cannot be zero. But why is this crucial for parallelism?
Simply put, a zero scalar would eliminate the presence of the vector entirely, leading to a zero vector, thereby losing the concept of direction altogether.
The importance of a non-zero scalar includes:
  • It keeps the vector intact, preserving its intrinsic properties.
  • It allows transformation and scaling without breaching the parallel condition.
  • Ensures that vectors remain valid entities with magnitude and direction, essential for establishing parallel relationships.
Thus, a non-zero scalar assures us that vectors, though transformed or scaled, adhere to the fundamental laws governing parallel vectors.

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Most popular questions from this chapter

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