When Cross Product Equals Dot Product: The Angle Between Vectors

The relationship between the cross product and dot product of two vectors is a fundamental concept in vector algebra. This article explores the scenario where the cross product of two vectors equals their dot product and derives the angle between them. This deep dive into vector analysis is valuable for students, researchers, and engineers working with vector calculus.

Introduction to Cross Product and Dot Product

The cross product and dot product are two important operations in vector algebra. The cross product of two vectors mathbf{A} and mathbf{B} , denoted as mathbf{A} times mathbf{B} , results in a vector that is perpendicular to both mathbf{A} and mathbf{B} , with its magnitude defined by mathbf{A} mathbf{B} sintheta mathbf{n}. Conversely, the dot product mathbf{A} cdot mathbf{B} is a scalar value equal to the product of the magnitudes of mathbf{A} and mathbf{B} and the cosine of the angle theta between them, i.e., mathbf{A} mathbf{B} costheta.

Condition: Cross Product Equals Dot Product

The problem at hand is to determine the angle between vectors mathbf{A} and mathbf{B} when the cross product equals the dot product:

mathbf{A} times mathbf{B} mathbf{A} cdot mathbf{B}

Let us express this condition mathematically:

mathbf{A} mathbf{B} sintheta mathbf{n} mathbf{A} mathbf{B} costheta

Analysis of the Conditions

To solve this equation, we must consider that the left side is a vector and the right side is a scalar. This implies two cases for the equality to hold:

Magnitude of the Cross Product is Zero

mathbf{A} mathbf{B} sintheta 0

This equation holds true if either mathbf{A} 0 , mathbf{B} 0 , or sintheta 0

- sintheta 0 occurs when theta 0^circ or theta 180^circ

When theta 0^circ, the vectors mathbf{A} and mathbf{B} are parallel. When theta 180^circ, the vectors are anti-parallel.

Magnitude of the Dot Product is Zero

mathbf{A} mathbf{B} costheta 0

This equation holds true if either mathbf{A} 0 , mathbf{B} 0 , or costheta 0

- costheta 0 when theta 90^circ

When theta 90^circ, the vectors are perpendicular.

Conclusion

The only angle that satisfies both conditions (i.e., the cross product being equal to the dot product) is when mathbf{A} and mathbf{B} are both zero vectors or when they are parallel/anti-parallel giving an angle of 0^circ or 180^circ. The angle theta cannot simultaneously be 90^circ as that would contradict the zero conditions.

Thus, theta between mathbf{A} and mathbf{B} can be:

- 0^circ (parallel)

- 180^circ (anti-parallel)

- Vectors are zero vectors

Alternative Approach to Deriving 45 Degrees

Another approach to solving this problem involves algebraic manipulation and the use of complex numbers. For the 2D case, the cross product is equivalent to the determinant of the 2x2 matrix formed by the vectors. Let's represent mathbf{A} amathbf{i} bmathbf{j} and mathbf{B} rmathbf{i} smathbf{j}. Squaring both sides of the equation and simplifying:

mathbf{A} cdot mathbf{B}^2 mathbf{A} times mathbf{B}^2

mathbf{a} mathbf{b} mathbf{r} mathbf{s} mathbf{a} mathbf{r} mathbf{b} mathbf{s} - mathbf{a} mathbf{b} mathbf{r} mathbf{s}

mathbf{a} mathbf{b} mathbf{r} mathbf{s} - mathbf{a} mathbf{b} mathbf{r} mathbf{s} 0

mathbf{a} mathbf{b} - mathbf{r} mathbf{s} mathbf{a} mathbf{b} 0

The first factor is characterized as:

mathbf{a} mathbf{b} - mathbf{r} mathbf{s} 0

frac{mathbf{r}}{mathbf{s}} frac{mathbf{a}}{mathbf{b}}

Using complex numbers for the direction of mathbf{B}, we consider the difference between their arguments:

arg(mathbf{a}-mathbf{b}i mathbf{a}mathbf{b}i) 45^circ

Thus, without directly using trig functions, we can conclude that the angle is 45^circ.