Dot Product vs Cross Product

The dot product, also known as the scalar product, is an algebraic operation that returns a single integer from two equal-length sequences of numbers (typically coordinate vectors). For example, the dot product of the cartesian coordinates of two vectors is commonly used in Euclidean geometry. Even though it is not the only inner product that may be defined on Euclidean space, it is frequently referred to as “the” inner product (or, more rarely, projection product) (see Inner product space for more).

The name “dot product” comes from the centred dot frequently used to denote this operation. The appropriate example, “scalar product”, stresses that the output is a scalar rather than a vector, as in three-dimensional space with the vector product.

The cross product, a×b (read “a cross b”), of two linearly independent vectors a and b, is a vector perpendicular to both a and b and so normal to the plane containing both. Mathematics, physics, engineering, and computer programming are just a few fields where they can be used. It is not to be confused with the dot item (projection product). The cross product of two vectors is 0 if they have the exact opposite directions (that is, they are not linearly independent) or if one of them has zero length. The area of a parallelogram with the vectors for sides equals the magnitude of the product; specifically, the magnitude of the product of two perpendicular vectors equals the product of their lengths.The cross product is anticommutative (a×b ≠ b×a). 

The product of the magnitude of the vectors and the cos of the angle between them is called a dot product. The magnitude of the vectors and the sine of the angle they subtend on each other form a cross product. A scalar quantity is the result of the dot product of the vectors.

Difference Between Dot Product and Cross Product

• In physics, engineering, and mathematics, the dot product and cross product have a variety of uses. The cross product, also known as a vector product, is a three-dimensional binary operation on two vectors. The outcome of the cross product is a vector that is perpendicular to both the multiplied vectors and normal to the plain.
• The dot product is an algebraic operation that takes two equal-length numbers and produces a single number. It’s calculated by multiplying the related elements and then combining the results.
• The dot product is denoted by “a. b” if the vectors are named “a” and “b.” This is equal to the angles’ cosine multiplied by the magnitudes. The cross product is denoted by “an X b” in vectors “a” and “b.” This is calculated by multiplying the magnitudes by the sine of the angles and then multiplying by “n,” a unit vector.
• The magnitude of a dot product is a maximum, whereas the magnitude of a cross product is zero. The metric of Euclidean space is used in both the dot product and the cross product. On the other hand, the cross product is reliant on choosing orientation.
• A dot product is commonly utilised when a vector must be projected onto another vector. The following are some instances of dot products:
• Calculating a point’s distance from a plane.
• Calculating the distance between two points on a line.
• Calculating a point’s projection.
• A cross product can be used for a variety of purposes, including:
• Calculating a point’s distance from a plane.
• The specular light is being calculated.

Vector Dot and Cross Products

A vector is a quantity defined not only by its magnitude but also by its direction. Vectors include things like velocity, force, acceleration, momentum, and so on.

There are two ways to multiply vectors:

• Dot product or scalar product
• Cross-product or vector product

Conclusion

The cross product, also known as the vector product, is a three-dimensional binary operation on two vectors. The dot product is an algebraic operation that takes two equal-length numbers and produces a single number. The cross product produces a perpendicular vector to both multiplied vectors and normal to the plane. Multiply the appropriate entries and then add the products to get the dot product. In a dot product, the magnitude is maximal, whereas it is zero in a cross product.

A dot product is commonly utilised when a vector must be projected onto another vector. The dot product is denoted as “a. b” if the vectors are designated “a” and “b” The cross product is denoted by “a X b” in vectors “a” and “b.”