Understanding the Angle Between Two Vectors When the Magnitude of Their Cross Product Equals the Absolute Value of Their Dot Product

In this article, we will explore the mathematical condition where the magnitude of the cross product of two vectors is equal to the absolute value of their dot product. This unique scenario will help us to determine the angle between these two vectors. We will start by defining the necessary vector operations, cross product, and dot product, before moving on to the detailed solution and conclusions.

Definitions of Vector Operations

To comprehend the problem, it is essential to clarify the definitions of the cross product and the dot product of vectors:

Cross Product

The cross product of two vectors (mathbf{a}) and (mathbf{b}) is a vector (mathbf{a} times mathbf{b}) with a magnitude given by:

(left|mathbf{a} times mathbf{b}right| left|mathbf{a}right| left|mathbf{b}right| sin theta)

where (theta) is the angle between the two vectors.

Dot Product

The dot product of two vectors (mathbf{a}) and (mathbf{b}) is a scalar given by:

(mathbf{a} cdot mathbf{b} left|mathbf{a}right| left|mathbf{b}right| cos theta)

where (theta) is the angle between the two vectors.

Given Condition and Analysis

Given that the magnitude of the cross product of two vectors is equal to the absolute value of their dot product, we have:

(left|mathbf{a} times mathbf{b}right| left|mathbf{a} cdot mathbf{b}right|)

Substituting the definitions of cross product and dot product, we get:

(left|mathbf{a}right| left|mathbf{b}right| sin theta left|mathbf{a}right| left|mathbf{b}right| cos theta)

Assuming (left|mathbf{a}right| left|mathbf{b}right| eq 0) (meaning both vectors are non-zero), we can divide both sides by (left|mathbf{a}right| left|mathbf{b}right|):

(sin theta cos theta)

This equation indicates two cases to consider:

Case 1: (cos theta geq 0)

In this case, (cos theta cos theta), and the equation simplifies to:

(sin theta cos theta)

This occurs when:

(tan theta 1 , Rightarrow , theta 45^circ , text{or} , frac{pi}{4} text{ radians})

Case 2: (cos theta

Here, (cos theta -cos theta), and the equation becomes:

(sin theta -cos theta)

This occurs when:

(tan theta -1 , Rightarrow , theta 135^circ , text{or} , frac{3pi}{4} text{ radians})

Summary

Thus, the angles (theta) that satisfy the condition (left|mathbf{a} times mathbf{b}right| left|mathbf{a} cdot mathbf{b}right|) are:

45deg; or (frac{pi}{4}) radians 135deg; or (frac{3pi}{4}) radians

To verify, let's assume vectors (mathbf{a}) and (mathbf{b}) with (theta) as the angle between them. The length of the cross product and dot product give:

(|mathbf{a} times mathbf{b}| |mathbf{a}| |mathbf{b}| sin theta) or (|mathbf{a} cdot mathbf{b}| |mathbf{a}| |mathbf{b}| cos theta)

Assuming nonzero vectors, we get:

Cosine and sine relationship holds, confirming the angles.

Conclusion

The keys to solving this problem include understanding the definitions and properties of dot and cross products, and applying trigonometric identities to simplify the given condition. The final angles are 45deg; and 135deg; or in radians, (frac{pi}{4}) and (frac{3pi}{4}).

Understanding these relationships is crucial in vector algebra and has applications in physics and engineering, particularly in analyzing vector quantities in 3D space.