To find a unit vector perpendicular to two vectors, calculate their cross product and then normalize the resulting vector.
Finding a vector that is orthogonal (perpendicular) to two other vectors is a fundamental operation in linear algebra and vector calculus. The primary tool for this in three-dimensional space is the cross product.
Understanding Orthogonality
As highlighted in vector mathematics, a vector V is perpendicular to a vector A if their dot product is zero (V ⋅ A = 0). Similarly, for V to be perpendicular to both A and B, it must satisfy both V ⋅ A = 0 and V ⋅ B = 0. This property is crucial when searching for such a vector.
The cross product inherently produces a vector that satisfies this condition for any two input vectors.
Using the Cross Product
The cross product of two vectors, say vector A and vector B, results in a new vector that is orthogonal to both A and B.
Let A = (a₁, a₂, a₃) and B = (b₁, b₂, b₃). The cross product A × B is calculated as follows:
A × B = (a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁)
This resulting vector C = A × B is perpendicular to the plane containing A and B.
Key Property:
The vector C = A × B is orthogonal to A and B. This means:
- C ⋅ A = 0
- C ⋅ B = 0
This aligns with the principle that an orthogonal vector V (which is C in this case) dotted with the original vectors (A and B) must yield zero.
Normalizing the Vector
The vector C = A × B is perpendicular to A and B, but it is not necessarily a unit vector. A unit vector has a magnitude (or length) of 1.
To convert any non-zero vector into a unit vector, you divide it by its magnitude. This process is called normalization.
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Calculate the magnitude of the vector C = (c₁, c₂, c₃):
|C| = √(c₁² + c₂² + c₃²) -
Divide the vector C by its magnitude:
Unit vector u = C / |C| = (c₁/|C|, c₂/|C|, c₃/|C|)
The resulting vector u is the unit vector perpendicular to both A and B.
Step-by-Step Guide
Here are the steps summarized to find a unit vector perpendicular to two given vectors A and B:
- Calculate the cross product C = A × B.
- This gives you a vector perpendicular to both A and B.
- Calculate the magnitude of the resulting vector C.
- |C| = √(c₁² + c₂² + c₃²)
- Normalize the vector C by dividing each component by its magnitude.
- Unit vector u = (c₁/|C|, c₂/|C|, c₃/|C|)
Important Note: The cross product A × B and B × A will result in vectors pointing in opposite directions (one "up", one "down" relative to the plane of A and B). Both are perpendicular to the original vectors. Therefore, there are two possible unit vectors perpendicular to the plane of A and B, which are u and -u.
Using the cross product method provides a direct way to find a vector V whose components (V₁, V₂, V₃) inherently satisfy the orthogonality conditions V ⋅ A = 0 and V ⋅ B = 0 without needing to solve a system of equations based on these dot products directly.
