Proving the Dot Product of a Vector with the Cross Product of Itself and Another Vector Equals Zero

Understanding vector algebra is crucial in various fields, including physics and engineering. One fundamental relationship in vector algebra is that the dot product of a vector with the cross product of itself and another vector is always zero. This article will explain why this is true and provide a proof for the statement.

Introduction to Cross Product

The cross product, denoted as A × B, is a binary operation between two vectors in a three-dimensional space. This operation results in a vector that is orthogonal to both original vectors. Mathematically, the cross product of two vectors A and B can be represented as:

A × B C, where C is a vector that is perpendicular to the plane containing A and B.

Dot Product with Orthogonal Vectors

The dot product, denoted as A · B, is a scalar obtained by multiplying corresponding entries of the two vectors and summing the results. In vector algebra, the dot product of two vectors is zero if and only if the vectors are orthogonal (perpendicular).

Key Proof

Given: A · (A × B) 0 This is because the vector resulting from the cross product A × B is orthogonal to both A and B. Therefore, the dot product of A with A × B is zero.

Step-by-Step Proof

Let us prove that A · (A × B) 0 step by step.

Express the vectors in component form: A (a, b, c) B (e, f, g) Compute the cross product A × B:

i???lab{A begin{pmatrix}a b cend{pmatrix} B begin{pmatrix}e f gend{pmatrix} A times B begin{pmatrix}bg - cf space; ce - ag space; af - beend{pmatrix} }

Compute the dot product of A with A × B:

i???lab{A cdot (A times B) a(bg - cf) b(ce - ag) c(af - be) abg - acf bce - bag acf - bce 0} }

Conclusion

The result of the dot product is zero because A × B is orthogonal to A. Therefore, the proof is complete, and we can conclude that:

A · (A × B) 0

This property of vector algebra is fundamental in understanding the behavior of vectors and is widely used in different fields such as physics, engineering, and mathematics.

Related Keywords

dot product cross product vector algebra

Further Reading and Resources

Dot Product on Wikipedia Cross Product on Wikipedia Cross Product Lecture Notes (PDF) Cross Product Explained (MathIsFun)