حاصل الضرب النقطي لمتجهين

The dot product combines two vectors and produces a scalar value.

Algebraically, it is calculated by multiplying the corresponding components of the vectors and adding the products. Geometrically, it equals the product of the magnitudes of both vectors and the cosine of the angle between them.

A practical application is optimizing the tilt of solar panels for maximum energy absorption. Sunlight is modeled as a three-dimensional direction vector.

Another vector represents the panel’s normal direction, perpendicular to its surface. Since sunlight travels toward the panel, its vector points inward. To align with the outward normal, the sunlight vector is reversed, and the dot product is calculated between the panel normal and the reversed vector.

This means multiplying the corresponding components and adding the results. Dividing this product by the vector magnitudes gives the cosine term that controls energy absorption by measuring how directly sunlight hits the panel.

When the angle is small, the cosine is large, meaning greater energy absorption.

At 90 degrees, the dot product becomes zero, and the panel receives no direct solar energy.