You can deduce that two vectors are parallel using specific mathematical relationships between them, based on whether one is a scaled version of the other, their cross product, or their dot product relative to their magnitudes.
Here are the key methods to determine if two vectors, say vector a and vector b, are parallel:
Methods to Determine Parallel Vectors
Based on fundamental vector properties, two vectors are considered parallel if they point in the same direction or in exactly opposite directions. This geometric property translates into several algebraic or vector operations.
1. Scalar Multiple Condition
Reference 1: If one vector is a scalar multiple of the other. i.e., a = kb, where 'k' is a scalar.
- This is the most intuitive definition. If vector a is simply a stretched, compressed, or reversed version of vector b, they are parallel.
- The scalar 'k' can be any real number.
- If k > 0, a and b are in the same direction.
- If k < 0, a and b are in opposite directions.
- If k = 0, then a = 0, the zero vector, which is conventionally considered parallel to all vectors.
2. Cross Product Condition
Reference 2: If their cross product is 0. i.e., a × b = 0.
- The magnitude of the cross product of two vectors is given by |a × b| = |a| |b| sin(θ), where θ is the angle between the vectors.
- For the cross product to be the zero vector (0), its magnitude must be 0.
- This occurs when |a| = 0, |b| = 0, or sin(θ) = 0.
- If neither vector is the zero vector (|a| ≠ 0 and |b| ≠ 0), then sin(θ) must be 0.
- sin(θ) = 0 when θ = 0° or θ = 180°.
- An angle of 0° means the vectors are in the same direction, and 180° means they are in opposite directions. Both cases signify that the vectors are parallel.
3. Dot Product and Magnitude Condition
Reference 3: If their dot product is equal to the product of their magnitudes. i.e., a · b = |a| |b|.
- The dot product of two vectors is given by a · b = |a| |b| cos(θ), where θ is the angle between the vectors.
- The condition a · b = |a| |b| implies that |a| |b| cos(θ) = |a| |b|.
- If neither vector is the zero vector, we can divide by |a| |b|, leaving cos(θ) = 1.
- cos(θ) = 1 when θ = 0°.
- An angle of 0° means the vectors point in the exact same direction, which is a specific case of being parallel.
- (Note: While this condition indicates parallelism in the same direction, the first two conditions cover both same and opposite directions.)
Summary Table
| Condition | Mathematical Expression | Parallelism Implied? | Notes |
|---|---|---|---|
| Scalar Multiple | a = kb | Yes | k can be any real number |
| Cross Product is Zero | a × b = 0 | Yes | Covers same and opposite directions |
| Dot Product = Product of Magnitudes | a · b = | a |
By verifying any of these conditions using the components of the vectors, you can mathematically deduce that two vectors are parallel.
