Why we use sin in cross product and cos in dot product. is there any scientific reason behind it? who uses its first? what is the inspiration from that it is derived?

The distinction between using sine in the cross product and cosine in the dot product is rooted in the geometric interpretations of these operations. Both the dot product and cross product are fundamental concepts in vector mathematics, and they serve different purposes in physics and engineering. Let’s break down the reasons behind this choice and explore the historical context as well.

Understanding the Dot Product

The dot product of two vectors is defined as:

A · B = |A| |B| cos(θ)

Here, θ is the angle between the two vectors A and B. The dot product measures how much one vector extends in the direction of another. The use of cosine is significant because:

• Projection: The dot product effectively projects one vector onto another. When you multiply the magnitudes of the vectors by the cosine of the angle, you are calculating the length of the projection of one vector onto the other.
• Angle Relationship: Cosine relates to the angle between the vectors. When the angle is 0 degrees (vectors are aligned), cos(0) = 1, giving the maximum value for the dot product. Conversely, when the vectors are perpendicular (90 degrees), cos(90) = 0, resulting in no contribution to the dot product.

Exploring the Cross Product

The cross product, on the other hand, is defined as:

A × B = |A| |B| sin(θ)

In this case, the sine function is used because:

• Area of the Parallelogram: The cross product gives a vector that is perpendicular to the plane formed by vectors A and B. The magnitude of this vector is equal to the area of the parallelogram defined by the two vectors, which is calculated using the sine of the angle between them.
• Directionality: The sine function captures the idea of the 'height' of the parallelogram when considering the angle. When the angle is 0 or 180 degrees, sin(0) = 0, indicating no area (the vectors are collinear). At 90 degrees, sin(90) = 1, giving the maximum area.

Historical Context and Inspiration

The mathematical foundations for these operations can be traced back to the work of mathematicians and physicists over centuries. The dot product concept can be linked to early studies in geometry and projection, while the cross product emerged from the need to understand three-dimensional space and rotational dynamics.

In the 19th century, mathematicians like William Rowan Hamilton and Josiah Willard Gibbs contributed significantly to vector calculus, formalizing these operations. The geometric interpretations of sine and cosine were already well established in trigonometry, which provided a natural framework for defining these products.

Applications Across Disciplines

Both the dot and cross products are widely used in various fields:

• Physics: In mechanics, the dot product is used to calculate work done, while the cross product is essential in determining torque and angular momentum.
• Engineering: Structural analysis often employs these products to understand forces and moments acting on structures.
• Computer Graphics: Both products are crucial in rendering scenes, calculating lighting, and determining orientations of objects.

In summary, the use of sine in the cross product and cosine in the dot product is deeply rooted in their geometric interpretations and applications. The historical development of these concepts reflects a blend of mathematical innovation and practical necessity, making them indispensable tools in science and engineering today.