Answer
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Work Step by Step
The cross product is a mathematical operation that takes two vectors as input and produces a new vector that is perpendicular to both of the input vectors. It has both geometric and physical interpretations.
Geometric interpretation:
1. Orthogonality: The cross product of two vectors is perpendicular to both of the input vectors. This property is useful in determining whether two vectors are parallel or perpendicular to each other.
2. Area: The magnitude of the cross product represents the area of the parallelogram formed by the two input vectors. The direction of the cross product determines the orientation of the parallelogram.
3. Right-hand rule: The direction of the cross product follows the right-hand rule. If you align your right-hand fingers along the first vector and then curl them towards the second vector, the direction your thumb points in is the direction of the cross product.
Physical interpretations:
1. Torque: In physics, the cross product is used to calculate torque. When a force is applied to an object at a distance from a pivot point, the torque is the cross product of the force vector and the displacement vector from the pivot point.
2. Magnetic fields: The magnetic field generated by a current-carrying wire can be determined using the cross product. The direction of the magnetic field is perpendicular to both the current direction and the displacement vector from the wire.
3. Angular momentum: The cross product is also used to calculate angular momentum. When an object is rotating, its angular momentum is the cross product of its moment of inertia and its angular velocity.
Example:
Let's consider two vectors A = (2, 3, 4) and B = (5, -1, 2). The cross product of A and B can be calculated as follows:
A x B = (3*2 - 4*(-1), 4*5 - 2*2, 2*(-1) - 3*5)
= (10, 16, -17)
Geometric interpretation: The cross product vector (10, 16, -17) is perpendicular to both A and B.
Physical interpretation: The cross product can be used to calculate torque, magnetic fields, or angular momentum in various physical scenarios.
