Proofs of the Cross-Product Formula: Understanding a × b |a||b|sinθ

Introduction to the Cross-Product Formula

In vector calculus, the cross product is a fundamental operation that results in a vector perpendicular to the two input vectors. The magnitude of the cross product is given by the formula |a × b| |a||b|sinθ where a and b are vectors, and θ is the angle between them. This formula is crucial for understanding geometric and physical applications involving vectors.

Geometric Interpretation of the Cross-Product Formula

The cross product can be interpreted geometrically as the area of the parallelogram formed by the vectors a and b. This area can be calculated as the base times the height, where the base is the magnitude of b and the height is the perpendicular distance from a to the line along b. The height can be expressed as |a|sinθ, leading to the formula:

A |b| × |a|sinθ

Since the area of the parallelogram is equal to the magnitude of the cross product, we have:

|a × b| |a||b|sinθ

Algebraic Proof Using Determinants

Algebraically, the cross product can be defined using the determinant of a matrix. For vectors a a1 a2 a3 and b b1 b2 b3, the cross product is computed as:

  cross(a, b)  begin{vmatrix}
    i  j  k 
    a_1  a_2  a_3 
    b_1  b_2  b_3 
  end{vmatrix}

Calculating the determinant gives:

a × b (a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1)

The magnitude of the cross product can be derived as follows:

|a × b| sqrt{(a_2 b_3 - a_3 b_2)^2 (a_3 b_1 - a_1 b_3)^2 (a_1 b_2 - a_2 b_1)^2}

Using the properties of the sine of the angle between the two vectors, this magnitude can be related back to the formula |a × b| |a||b|sinθ.

Conclusion

Both the geometric interpretation and the algebraic formulation converge to the same conclusion, confirming that the cross product magnitude formula is valid:

|a × b| |a||b|sinθ

This establishes the validity of the cross-product formula and highlights its importance in various mathematical and physical applications.