Final answer:
In order to find a unit vector perpendicular to both vectors ~a and ~b, calculate the cross product ~a × ~b, which gives a vector perpendicular to both. To make it a unit vector, divide this cross product by its magnitude.
Step-by-step explanation:
In order to find a unit vector perpendicular to both vectors ~a and ~b, we use the vector product, also known as the cross product.
The cross product ~a × ~b yields a vector that is perpendicular to both ~a and ~b.
The magnitude of this vector is given by |~a × ~b| = |~a||~b|sinθ, where θ is the angle between vectors ~a and ~b.
To make this a unit vector, we must divide the cross product by its magnitude.
The formula for the cross product is as follows:
Vector product (cross product): ~a × ~b produces a vector that is perpendicular to both vectors ~a and ~b.
Magnitude: The magnitude of the cross product is |~a||~b|sinθ.
Direction: The direction of the cross product is determined by the right-hand rule, which states that if you curl the fingers of your right hand from ~a to ~b, your thumb points in the direction of ~a × ~b.
Unit vector: To find the unit vector, divide the vector product by its magnitude: â = (~a × ~b)/|~a × ~b|.