When Do the Magnitudes of the Cross Product Equal the Dot Product of Two Vectors?

Understanding the relationship between the dot product and cross product of two vectors is fundamental in mathematics and physics. While the dot product yields a scalar value representing the projection of one vector onto another, the cross product results in a vector perpendicular to the plane containing the two vectors. In this article, we explore the conditions under which the magnitudes of the cross product and the dot product of two vectors are numerically equal. We will delve into the mathematical derivations and provide a clear answer to this intriguing question.

Defining the Problem

Suppose we have two vectors a and b. The cross product and dot product of these vectors can be expressed as:

Dot Product: a cdot mathit{b} |mathit{a}||mathit{b}|costheta Cross Product: a times mathit{b} |mathit{a}||mathit{b}|sintheta(mathit{u}, where is a unit vector perpendicular to both a and b.

Our goal is to find the angle theta such that the magnitude of the cross product equals the magnitude of the dot product, i.e.,

|a times mathit{b}| |mathit{a} cdot mathit{b}|

Deriving the Solution

Assuming both vectors are non-zero, we can start by considering the magnitudes of the vectors. The magnitudes of the dot and cross products are given by:

Dot Product: |mathit{a}||mathit{b}|costheta Cross Product: |mathit{a}||mathit{b}|sintheta

To find the angle theta such that these magnitudes are equal, we set up the following equation:

|mathit{a}||mathit{b}|sintheta |mathit{a}||mathit{b}|costheta

Dividing both sides by the non-zero product |mathit{a}||mathit{b}|, we obtain:

sintheta costheta

Dividing both sides by costheta (assuming costheta eq 0), we get:

tantheta 1

The general solution to this equation is:

theta 45^circ 180^circ k quad text{for } k in mathbb{Z}

Therefore, the angles between the vectors a and b such that their cross product magnitude is equal to their dot product magnitude are:

-45^circ 225^circ -135^circ 405^circ

Converting these to standard angles, we get:

-45^circ 225^circ -135^circ equiv 225^circ 405^circ - 360^circ 45^circ

Thus, the solutions within the standard range of angles are:

45^circ 225^circ

Conclusion

The angle between two vectors such that their cross product magnitude is equal to their dot product magnitude is 45^circ and 225^circ. This discovery has significant implications in fields such as vector calculus, physics, and engineering.