To find a vector perpendicular to two given vectors, you use the cross product.
The most common and direct way to find a vector that is perpendicular to two other vectors in three-dimensional space is by calculating their cross product.
As we all know, the magnitude of the cross product of two vectors is equal to the product of their magnitudes and the sine of the angle between them. More importantly, the cross product produces a vector that is inherently perpendicular to both original vectors. This is because, conceptually, the cross product relates to the area of the parallelogram formed by the two vectors, and the area vector of any surface is defined in a direction perpendicular to that surface.
Let's say you have two vectors, A and B. Their cross product, denoted by A × B, results in a new vector that is orthogonal (perpendicular) to both A and B.
The Formula for the Cross Product
If your two vectors are in Cartesian coordinates:
A = A₁i + A₂j + A₃k
B = B₁i + B₂j + B₃k
The cross product A × B is calculated using the determinant of a matrix:
A × B = |
| i | j | k |
| A₁ | A₂ | A₃ |
| B₁ | B₂ | B₃ |
|
Expanding the determinant gives you the components of the resulting perpendicular vector:
A × B = (A₂B₃ - A₃B₂) i - (A₁B₃ - A₃B₁) j + (A₁B₂ - A₂B₁) k
This resulting vector is perpendicular to both A and B.
Steps to Calculate the Cross Product
- Identify the components of the two vectors (A₁,A₂, A₃ and B₁,B₂, B₃).
- Set up the determinant matrix with the unit vectors (i, j, k) in the first row, the components of the first vector in the second row, and the components of the second vector in the third row.
- Calculate the determinant. This involves:
- Multiplying i by the determinant of the 2x2 matrix formed by removing the first row and first column.
- Subtracting j multiplied by the determinant of the 2x2 matrix formed by removing the first row and second column.
- Adding k multiplied by the determinant of the 2x2 matrix formed by removing the first row and third column.
- The result is a vector with the components calculated in the previous step.
Example
Let's find a vector perpendicular to A = 2i + 3j + 1k and B = -1i + 5j + 2k.
Using the formula:
A × B = ((3)(2) - (1)(5)) i - ((2)(2) - (1)(-1)) j + ((2)(5) - (3)(-1)) k
A × B = (6 - 5) i - (4 - (-1)) j + (10 - (-3)) k
A × B = 1i - 5j + 13k
The vector i - 5j + 13k is perpendicular to both A and B.
Important Considerations
- The cross product only applies to vectors in three-dimensional space.
- The order matters: B × A is the negative of A × B, meaning it points in the opposite direction but is still perpendicular to both vectors.
- If the two vectors are parallel, their cross product is the zero vector (0), as the angle between them is 0° or 180°, and sin(0°) = sin(180°) = 0. The zero vector is conventionally considered perpendicular to all vectors.
In summary, the cross product is the definitive method for finding a vector orthogonal to two given vectors in 3D space.
