Understanding the Angle Between Vectors A and B Given A·B A×B

When considering vector multiplication, it is important to understand the differences between the dot product and the cross product. This article will explore the scenario where a vector A multiplied by a vector B is equal to their dot product:

Definitions and Notations

First, let's define the notations:

A and B represent vectors The magnitude of vector A and B are denoted as A and B θ represents the angle between vectors A and B where 0° θ 180°

Magnitude of the Cross Product

The cross product of vectors A and B is defined as:

A times; B AB sin(θ)

Magnitude of the Dot Product

The dot product of vectors A and B is given by:

A · B AB cos(θ)

Equating the Cross Product and Dot Product

Given the condition A times; B A · B, we can set the equations equal to each other:

AB sin(θ) AB cos(θ)

Assuming A and B are non-zero, we can divide both sides by AB:

sin(θ) cos(θ)

Solving for θ

The equation sin(θ) cos(θ) implies:

tan(θ) 1

The angle θ that satisfies this condition is:

θ 45° or θ 225°

Therefore, the angle between vectors A and B can be either 45° or 225°.

Reconciling A · B and A × B

Considering the vector and scalar products, we see:

A · B produces a scalar (magnitude only).

A times; B produces a vector (magnitude and direction).

Given the condition A · B A times; B leads to:

AB cos(θ) AB sin(θ)

This simplifies to:

1 tan(θ)

Thus, θ 45°

The angle between A and B, given the condition, is 45°.

Conclusion

In summary, the angle between vectors A and B is 45 degrees when their dot product equals their cross product. This calculation involves understanding the concepts of vector multiplication, specifically the dot and cross products, and how they interrelate under specific conditions.