Final answer:
The cross product of two vectors results in a vector that is perpendicular to both and is computed using the magnitudes of the vectors, the sine of the angle between them, and the cyclic order of unit vectors. The resulting vector is constructed by algebraically combining the product of the components following a specified formula.
Step-by-step explanation:
The Vector Product (Cross Product)
The cross product of two vectors ℝ and Β is defined as a vector that is perpendicular to both vectors ℝ and Β. The cross product, denoted by ℝ × Β, results in a third vector Č. The computation of the cross product takes into account the magnitudes of the vectors, the sine of the angle between them, and the cyclic order of unit vectors.
For vectors ℝ = Axī + Ayĸ + AzÅ and Β = Bxī + Byĸ + BzÅ, the cross product can be obtained algebraically by using the following vector equation:
Č = ℝ × Β = (Ay Bz − Az By)ī + (Az Bx − Ax Bz)ĸ + (Ax By − Ay Bx)Å.
It is important to sketch the vectors ℝ and Β, as well as their cross product vector Č, to visually understand their orientation in space. The cross product vector Č will be perpendicular to the plane formed by vectors ℝ and Β and will follow the right-hand rule for direction.