Introduction to Vector Multiplication

Vector multiplication can be divided into two categories. A vector has both a magnitude and a direction, and as a result, the dot product of two vectors and the cross product of two vectors are the two methods of multiplying vectors. Because the resultant value is a scalar quantity, the dot product of two vectors is often referred to as the scalar product of two vectors. The cross product is referred to as the vector product because the output is a vector that is perpendicular to the two vectors that were used to compute it. 

Let us study about the two-way multiplication of vectors, including the working rule, properties, applications, and examples of this type of multiplication.

 Best way to do vector multiplication

A vector has both a magnitude and a direction associated with it. Dot product and cross product are two methods of multiplying two or more vectors. Please allow us to learn more about each of the vector multiplication operations.

Dot Product

The dot product of vectors, commonly known as the scalar product of vectors, is a mathematical operation on vectors. The scalar value produced by the dot product of the vectors is the product of the vectors. When two vectors are multiplied together, they form a dot product that is equal to the product of their magnitudes, plus the sine of the angle between the two vectors. The consequent of the dot product of two vectors is located in the same plane as the two vectors that were used to compute it. The dot product can be either a positive real number or a negative real number in the domain of the real numbers.

Let a and b represent two non-zero vectors, and let represent the angle between the vectors. The scalar product, often known as the dot product, is symbolised by the symbol a.b, which is defined as:

 The relationship between the variables a and b is | a | | b | cosθ.

,where |a| is the magnitude of a, |b| is the magnitude of b, and the angle between them is represented by | a |, | b |, and θ is the angle between them. 

Cross Product 

A Cross Product is sometimes referred to as a Vector Product in some circles. This type of vector multiplication is conducted between two vectors of varying nature or sorts and is known as the cross product. Whenever two vectors are multiplied with each other, and the multiplication is also a vector quantity, the resulting vector is referred to as the cross product of two vectors, or the vector product, respectively. When the two supplied vectors are combined, the resultant vector is perpendicular to the plane containing the two vectors.

We may further appreciate this by considering the following example: if we have two vectors lying in the X-Y plane, their cross product will result in a resultant vector pointing in the direction of the Z-axis, which is perpendicular to the X-Y plane. The arrow symbol is used to connect the two initial vectors together. The following diagram illustrates the vector multiplication or the cross-product of two vectors.

a x b=c

There are two vectors in this equation, and the resultant vector is represented by the letter c. Suppose that a and b form an angle of 90 degrees, and n is the unit vector perpendicular to the plane containing both a and b The cross product of the two vectors can be calculated using the following equation:

a x b=|a| |b|sinθ

Conclusion

Vector multiplication can be divided into two categories. A vector has both a magnitude and a direction, and as a result, the dot product of two vectors and the cross product of two vectors are the two methods of multiplying vectors. Because the resultant value is a scalar quantity, the dot product of two vectors is often referred to as the scalar product of two vectors. The cross product is referred to as the vector product because the output is a vector that is perpendicular to the two vectors that were used to compute it. A vector has both a magnitude and a direction associated with it. Dot product and cross product are two methods of multiplying two or more vectors. The dot product of vectors, commonly known as the scalar product of vectors, is a mathematical operation on vectors. The scalar value produced by the dot product of the vectors is the product of the vectors.