Two vectors are parallel to each other when they point in the same or exactly opposite directions.
According to the reference, two vectors are parallel when the angle between them is either 0° (the vectors point in the same direction) or 180° (the vectors point in opposite directions). This fundamental property defines vector parallelism, regardless of their magnitudes.
Understanding Vector Parallelism
Vector parallelism describes a specific geometric relationship between two vectors. It essentially means they lie on the same line or on parallel lines. This condition is met under two primary scenarios:
- Same Direction (0° Angle): If two vectors are parallel and point in the same direction, the angle between them is 0 degrees. Imagine two arrows starting from the same point and heading towards the same target; they are parallel and aligned.
- Opposite Directions (180° Angle): If two vectors are parallel but point in opposite directions, the angle between them is 180 degrees. Think of two arrows starting from the same point but heading away from each other along a straight line; they are parallel but opposing.
How to Determine if Vectors Are Parallel
Beyond the angle definition, there are practical ways to check for vector parallelism:
- Scalar Multiple: The most common method in linear algebra is checking if one vector is a scalar multiple of the other. If vector v is parallel to vector w, then v = *kw, where k* is a non-zero scalar (a real number).
- If k > 0, the vectors point in the same direction (angle is 0°).
- If k < 0, the vectors point in opposite directions (angle is 180°).
- Cross Product (in 3D): For non-zero vectors v and w in three dimensions, they are parallel if and only if their cross product is the zero vector: v × w = 0. This is because the magnitude of the cross product is |v||w|sin(θ), and sin(θ) is zero when θ is 0° or 180°.
- Slope (in 2D): In two dimensions, two non-vertical vectors are parallel if they have the same slope. Vertical vectors are parallel to each other.
Practical Examples
Let's look at some examples using the scalar multiple concept:
- Example 1: Same Direction
- Vector u = (2, 4)
- Vector v = (4, 8)
- Here, v = 2 u. Since the scalar k* = 2 is positive, u and v are parallel and point in the same direction.
- Example 2: Opposite Direction
- Vector a = (3, -6)
- Vector b = (-1, 2)
- Here, a = -3 b. Since the scalar k* = -3 is negative, a and b are parallel and point in opposite directions.
- Example 3: Not Parallel
- Vector p = (1, 2)
- Vector q = (3, 5)
- There is no single scalar k such that q = *kp. If we check the first component, 3 = k 1 implies k = 3. But for the second component, 5 = k 2 implies k = 2.5. Since k* is not consistent, p and q are not parallel.
Summary of Parallel Vector Properties
| Property | Description |
|---|---|
| Angle | 0° (same direction) or 180° (opposite direction) |
| Scalar Multiple | One vector is a non-zero scalar multiple of the other (v = *kw, k* ≠ 0) |
| Cross Product | The cross product is the zero vector (v × w = 0 for non-zero v, w) |
| Slope (2D) | Same slope (or both vertical) |
Understanding vector parallelism is crucial in physics, engineering, and computer graphics for describing forces, velocities, directions, and more.
