Understanding the Vector Product of Unit Vectors: A 60° Angle

When dealing with vectors in mathematics and physics, the vector product (cross product) is a fundamental concept. This article explains the magnitude of the vector product of two unit vectors that form a 60° angle with each other. Understanding this concept is essential for fields such as engineering, physics, and computer graphics.

Definition of Unit Vectors and Vector Product

A unit vector is a vector that has a magnitude of one and is often used in vector operations. The vector product, also known as the cross product, of two vectors (vec{A}) and (vec{B}) is defined by the formula:

[vec{A} times vec{B} |vec{A}| |vec{B}| sin(theta) hat{n}]

(|vec{A}|) and (|vec{B}|) are the magnitudes of the vectors. (theta) is the angle between the two vectors. (hat{n}) is a unit vector perpendicular to both (vec{A}) and (vec{B}).

Calculating the Vector Product for Unit Vectors at 60°

Let us consider two unit vectors (vec{OA}) and (vec{OB}) such that (vec{OA} vec{OB} 1) unit. When these vectors form a 60° angle with each other, we can calculate their vector product as follows:

Determine the magnitudes of the vectors: Both (vec{OA}) and (vec{OB}) have a magnitude of 1. Identify the angle between the vectors: (theta 60°). Calculate the sine of the angle: (sin(60°) frac{sqrt{3}}{2}). Substitute the values into the formula: [|vec{OA} times vec{OB}| |vec{OA}| |vec{OB}| sin(60°) 1 cdot 1 cdot frac{sqrt{3}}{2} frac{sqrt{3}}{2}]

The magnitude of the vector product is (frac{sqrt{3}}{2}), which is approximately 0.866. This result indicates that the vector product produces a vector perpendicular to both (vec{OA}) and (vec{OB}) with a magnitude of (frac{sqrt{3}}{2}).

Explanation of the Result

The vector product of two unit vectors at a 60° angle forms a vector that is perpendicular to the plane containing (vec{OA}) and (vec{OB}). This vector's magnitude is solely a function of the sine of the angle between the original vectors. Thus, we have:

[|vec{A} times vec{B}| frac{sqrt{3}}{2}]

This result can be broken down further using the dot product (for a different angle, also known as the cosine of the angle) and illustrated using the geometric interpretation of the vectors and their orthogonal relationship in three-dimensional space.

Visualizing the Concept

Consider a 3D coordinate system where each unit vector lies along the axes. For simplicity, let (vec{OA}) lie along the positive x-axis and (vec{OB}) lie in the xy-plane forming a 60° angle with the x-axis. The vector product (vec{A} times vec{B}) will be a vector in the z-direction with a magnitude of (frac{sqrt{3}}{2}).

Note: The cross product is crucial for determining torque, magnetic fields, and other cross-sectional calculations in physics and engineering.

Conclusion

The magnitude of the vector product of two unit vectors forming a 60° angle is (frac{sqrt{3}}{2}), or approximately 0.866. Understanding this concept is vital for advanced mathematical and physical applications. By mastering the vector product, one can explore a variety of fields that rely on vector operations to describe and analyze complex systems.