Question

State the definition of parallel vectors.

Short answer

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

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 \( \mathbf{b} = \lambda \mathbf{a} \).

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, \( \mathbf{a} = 2\mathbf{b} \).

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.