Parallel vectors

Definition

Parallel vectors are vectors that have the same or exactly opposite direction. They can be scaled versions of each other, meaning one vector is a scalar multiple of the other.

5 Must Know Facts For Your Next Test

1. Two vectors $\mathbf{A}$ and $\mathbf{B}$ are parallel if there exists a scalar $k$ such that $\mathbf{A} = k\mathbf{B}$.
2. If two vectors are parallel, their cross product is zero: $\mathbf{A} \times \mathbf{B} = \mathbf{0}$.
3. The dot product of parallel vectors is equal to the product of their magnitudes: $\mathbf{A} \cdot \mathbf{B} = |\mathbf{A}| |\mathbf{B}|$ if they are in the same direction and $-\|\mathbf{A}\| \|\mathbf{B}\|$ if they are in opposite directions.
4. Parallel vectors have equal or directly proportional direction cosines.
5. In physics problems involving forces, parallel vectors often imply that the forces act along the same line of action.

Review Questions

• What condition must be satisfied for two vectors to be considered parallel?
• How can you determine if two given vectors are parallel using their cross product?
• What will be the dot product result if two vectors are parallel but in opposite directions?

Related terms

Scalar Multiple: A scalar multiple of a vector is obtained by multiplying a vector by a scalar, changing its magnitude but not its direction.

$\textit{\textbf{{Cross Product}}}$: $\textit{\textbf{{The cross product}}}$ $$ (\\vec\textit{\textbf{{u}}}} \times \vec\textit{\textbf{{v}}}) $$ results in a vector that is perpendicular to both original vectors with magnitude equal to the area of the parallelogram formed by them.

$\textit{\textbf{{Dot Product}}}$: $\textit{\textbf{{The dot product}}}$ $(\vec\textit{\textbf{{u}}}\cdot \vec\textit{\textbf{{v}}})$ measures how much one vector extends in the direction of another and results in a scalar.