Dot Product vs Cross Product
Dot product and cross product are two mathematical operations used in vector algebra, which is a very important field in algebra. These concepts are widely used in fields such as electromagnetic field theory, quantum mechanics, classical mechanics, relativity and many other fields in physics and mathematics. In this article, we are going to discuss what dot product and cross product are, their definitions and applications, some basic relationships concerning dot product and cross product, and finally the difference between dot product and cross product.
Dot product, also known as the scalar product, is a mathematical operator used in vector algebra. The dot product of two vectors A and B is defined as |A||B| Cos (θ), where θ is the angle measured between A and B. It can be obviously seen that the value of the dot product is a scalar value; therefore, the dot product is also known as the scalar product. The dot product yields a maximum value when the two vectors are parallel to each other. The minimum value of the dot product is when the two vectors are antiparallel. The dot product can be used also to take the projection of a vector in a given direction; for this, the second vector must be the unit vector in the desired direction. The dot product is also very useful in taking area integrals for Gauss’ theorem. It also plays a role in the differential operation divergence. Dot product is also used to calculate the work done in a force field.
Cross product, also known as the vector product, is a mathematical operation used in vector algebra. The cross product between the two vectors A and B are defined as |A||B| Sin (θ) N, where θ is the angle between A and B, and N is the unit normal vector to the plane that contain A and B. The direction of N is determined by the right handed screw rule from the direction of A to B. The modulus of the dot product is a maximum when the angle between A and B are 90 degrees (π/2 radians). The cross product is used to calculate the curl of a vector field. It is also used to calculate angular momentum, angular velocity and other properties of angular motion.
What is the difference between Dot Product and Cross Product?
• Dot product yields a scalar value, whereas the cross product yields a vector.
• The cross product takes the maximum value when the two vectors are perpendicular to each other, but the dot product takes the maximum when the two vectors are parallel to each other.
• Dot product is used to calculate the divergence of a vector field, but the cross product is used to calculate the curl of the vector field.