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Chapter 9: Problem 84

State the definition of parallel vectors.

Short Answer

Expert verified
Parallel vectors are scalar multiples of each other, meaning one vector is a stretch of the other. They have the same or opposite direction but can have different magnitudes.

Step by step solution

01

Definition

Vectors are said to be parallel if they are scalar multiples of each other. This implies that one vector is a stretch of the other. That is, for two vectors \( \mathbf{a} \) and \( \mathbf{b} \), they are parallel if there exists a real number \( \lambda \) such that \( \mathbf{a} = \lambda \mathbf{b} \) or \( \mathbf{b} = \lambda \mathbf{a} \).
02

Illustration

For example, consider two vectors \( \mathbf{a} = 2\mathbf{i} + 2\mathbf{j} \) and \( \mathbf{b} = \mathbf{i} + \mathbf{j} \). These vectors are parallel because \( \mathbf{a} \) is 2 times \( \mathbf{b} \), or shortly, \( \mathbf{a} = 2\mathbf{b} \).
03

Property

An important property to remember is that parallel vectors can have different magnitudes (or lengths), but they always have the same or opposite direction.

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