A vector can be perpendicular to two other vectors simultaneously by being the result of their cross product.
Understanding how a single vector achieves perpendicularity to two distinct vectors requires exploring the concept of the cross product in vector algebra. This operation is specifically designed to produce a vector that satisfies this condition.
The Cross Product: The Solution
The primary method to find a vector that is perpendicular to two given vectors is by calculating their cross product.
Let's say you have two vectors, A and B. The cross product, denoted by A × B, results in a new vector, let's call it C.
A × B = C
Vector C is perpendicular to both vector A and vector B. This property is fundamental to the definition and utility of the cross product.
Properties of the Cross Product
- Direction: The resulting vector C is perpendicular to the plane containing both A and B. Its specific direction is determined by the right-hand rule.
- Magnitude: The magnitude of the cross product |A × B| is given by |A| |B| sin(θ), where |A| and |B| are the magnitudes of vectors A and B, and θ is the angle between them (0° ≤ θ ≤ 180°). As stated in the reference, this magnitude is equal to the product of their magnitudes and the sine of the angle between them.
- Zero Result: If vectors A and B are parallel or antiparallel (θ = 0° or 180°), their cross product is the zero vector (0), as sin(0°) = sin(180°) = 0. The zero vector is considered perpendicular to all vectors.
Why the Cross Product Works
The reason the cross product yields a vector perpendicular to both inputs is deeply rooted in its geometric interpretation.
As mentioned in the reference: "The cross product produces a vector that is perpendicular to both vectors because the area vector of any surface is defined in a direction perpendicular to that surface."
Think of vectors A and B as forming the sides of a parallelogram. The magnitude |A × B| represents the area of this parallelogram. The cross product vector C is defined to point in the direction perpendicular to the plane of this parallelogram. This perpendicular direction is inherently perpendicular to both sides A and B that form the parallelogram.
It's like defining the "up" direction from a flat surface – it's the direction perpendicular to that surface, and thus perpendicular to any direction on that surface.
Applications
This ability to find a vector perpendicular to two others has numerous applications in physics, engineering, and computer graphics, including:
- Calculating Torque: Torque (τ) is the rotational equivalent of force and is calculated as the cross product of the position vector (r) and the force vector (F): τ = r × F. The torque vector is perpendicular to both the position vector and the force vector.
- Finding the Normal Vector to a Plane: Given two non-parallel vectors lying in a plane, their cross product yields a vector perpendicular to that plane (a normal vector). This is crucial in 3D geometry and calculus.
- Calculating Angular Momentum: Angular momentum (L) is calculated as the cross product of the position vector (r) and the linear momentum vector (p): L = r × p.
- Determining Magnetic Force: The magnetic force (F_B) on a moving charge is given by F_B = q(v × B), where q is the charge, v is the velocity, and B is the magnetic field. The force is perpendicular to both the velocity and the magnetic field vectors.
Summary
In conclusion, a vector can be perpendicular to two other vectors specifically by being the result of their cross product. This geometric property, linked to the concept of an area vector being perpendicular to its surface, ensures that the resulting vector is orthogonal to both input vectors, making the cross product an indispensable tool in various scientific and engineering fields.
