## Vectors and the Dot Product

Vectors are essential mathematical tools used in various fields like physics, engineering, and economics. Vectors are quantities that contain both magnitude and direction.

The magnitude of a vector is the length or size of the vector, while direction indicates the orientation of the vector. In this article, we focus on the dot product of vectors, which is a mathematical operation that takes two vectors and returns a scalar value.

We will also look at the vdot( ) function from the Numpy library, which computes the dot product between arrays.

### 1) Definition of Vectors

Before delving into the details of the dot product, it is essential to understand what vectors are. A vector is a quantity that has both magnitude and direction.

Vectors are denoted by an arrow, with the direction of the arrow representing the direction of the vector, and the length of the arrow representing the magnitude of the vector. Vectors can be represented using coordinates or as column or row matrices.

The magnitude of a vector can be found using the Pythagorean theorem. For example, if we have a vector (3, 4), the magnitude of the vector is $sqrt{3^2 + 4^2}$, which is 5.

The direction of a vector can be measured in degrees or radians from a reference axis.to Vector Dot Product

### 2) Vector Dot Product

The dot product, also known as scalar product, is a mathematical operation between two vectors. The dot product of two vectors gives a scalar value, which is the product of the magnitudes of the vectors and the cosine of the angle between them.

The dot product is denoted using a dot (.) between the two vectors. For example, given vectors $vec{a}$ and $vec{b}$, the dot product can be computed as $vec{a}cdot vec{b} = |vec{a}||vec{b}|costheta$, where $theta$ is the angle between the vectors.

The dot product has several applications in mathematics and physics. For instance, the work done by a force acting on an object can be computed using the dot product.

Similarly, the projection of a vector onto another vector or plane can be obtained using the dot product.

### 3) Difference between dot( ) and vdot( ) functions

The Numpy library provides two functions for computing the dot product of two vectors. These functions are dot( ) and vdot( ).

The dot( ) function computes the dot product of two arrays, while the vdot( ) function computes the dot product of two N-dimensional arrays. The dot( ) function computes the dot product by taking the element-wise product of the two arrays and then summing the resulting values.

For example, let us take two arrays a = [1,2,3] and b = [4,5,6]. The dot product can be computed as dot(a, b) = 1*4 + 2*5 + 3*6 = 32.

On the other hand, the vdot( ) function computes the dot product by first flattening the arrays, taking the complex conjugate of the second array, and then performing the dot product of the resulting arrays. The complex conjugate technique ensures that the dot product of complex vectors is computed correctly.

For instance, consider two complex vectors a = [2+3j, 4-5j] and b = [1+2j, 3+7j]. The dot product can be computed as vdot(a, b) = (2-3j)*(1-2j) + (4+5j)*(3-7j) = -4-28j.

The vdot( ) function also works for N-dimensional arrays, where the dot product is computed for the last dimension of the arrays. For higher-dimensional arrays, the arrays are flattened before computing the dot product.

### 4) Syntax of vdot( ) Function

The vdot( ) function can be used to compute the dot product of two N-dimensional arrays. The syntax for the vdot( ) function is as follows:

vdot(a, b)

where a and b are N-dimensional arrays representing the vectors whose dot product is to be computed.

### 5) Inputs for vdot( ) Function

The vdot( ) function takes two inputs, which are N-dimensional arrays. Each array must have the same number of elements along the last dimension, where the dot product is computed.

The arrays can be complex numbers or real numbers.

## 3) Calculating Vector Dot Product for N-Dimensional Arrays

Vector dot product is a mathematical operation that finds the scalar value of two vectors, which is their product of magnitudes and cosine of the angle between them. Numpy library provides two functions to perform the dot product of two vectors, namely dot( ) function and vdot( ) function.

In this section, we will explore the calculation of the vector dot product of n-dimensional arrays using an example using two-dimensional arrays.

### Example using two-dimensional arrays

Let us consider two matrices, A and B, represented as two-dimensional arrays, where the number of columns of A is equal to the number of rows of B. Suppose we have A = [[1, 2], [3, 4], [5, 6]] and B = [[7, 8, 9], [10, 11, 12]].

To compute the dot product using the dot( ) function, we can use the following syntax –

`result = np.dot(A, B)`

This code will give us a result where the number of rows will be the same as the number of rows of A and the number of columns equal to the number of columns of B. Hence, in this case, the result will be a (3, 3) matrix.

### Computation of vector dot product

To calculate the vector dot product, we need to take the dot product of corresponding elements from the two vectors and add the results. In the above example, let us assume A = [1, 3, 5] and B = [2, 4, 6].

The dot product of A and B will be A.B = (1*2) + (3*4) + (5*6) = 44. Hence the scalar product of A and B is 44.

In general, to perform dot product for two n-dimensional arrays, each having ‘k’ dimensions, the operation is carried out for the last dimension. For example, to find the dot product for two arrays A and B of shape (2, 3, 5), the dot product will be computed as below:

`result = np.sum(A * B, axis=-1) # axis=-1 means along the last dimension`

Here, the axis parameter specifies along which dimension the dot product needs to be calculated.

The output of this operation will be a matrix of shape (2, 3), indicating that the dot product was computed along the last dimension.

## 4) Calculating Vector Dot Product for Complex Numbers

Complex numbers are represented in a+bi format, where ‘a’ and ‘b’ are real numbers and ‘i’ represents the imaginary unit. The vdot( ) function in the Numpy library provides an efficient way to calculate the dot product of two complex vectors while utilizing the complex conjugate technique.

### Example using complex numbers

Let us consider two complex vectors, a and b, where a = [3+4j, 2-5j] and b = [1+6j, 4-3j]. To compute the vector dot product using the vdot( ) function, we can use the following syntax –

`result = np.vdot(a, b)`

The output of this operation will be (19+29j), giving us the scalar value for the dot product of the two complex vectors.

### Utilization of complex conjugate technique in vdot( ) function

The vdot( ) function finds the dot product by taking the complex conjugate of the second vector and then performing the dot product of flattened arrays. This technique ensures that the dot product of complex vectors is computed correctly.

The complex conjugate of a complex number a+bi is given by a-bi. To understand this technique let us consider the example of complex numbers, a and b.

In a vector dot product, the complex conjugate of the second vector, b, is taken to obtain the dot product. Thus, the vdot( ) function computes the dot product as shown below:

`result = a[0]*b[0].conjugate() + a[1]*b[1].conjugate()`

In the above equation, the dot product is computed using the complex conjugate of the second element of the vector b.

The operation then sums these products to give the resultant scalar value of the dot product.

In general, the complex conjugate technique is used when performing dot product operations with complex vectors to address the addition of the imaginary part of the resultant scalar value.

## Conclusion

The vector dot product is a mathematical operation performed on two vectors that returns a scalar value. Numpy library provides two functions to perform the dot product of two vectors, the dot( ) function, and vdot( ) function.

We can compute the dot product of n-dimensional arrays by concatenating the last dimension. Further, the vdot( ) function also helps in computing the dot product for complex numbers accurately by employing the complex conjugate technique.

In this article, we looked at examples and syntax for performing the dot product operation on arrays and vectors.

## Conclusion

In this article, we introduced the concept of vectors and the vector dot product, which is a mathematical operation used to calculate the scalar product of two vectors. The dot product has several applications in fields such as physics and engineering and can be computed using two functions from the Numpy library – the dot( ) function and the vdot( ) function.

We discussed the definition of vectors in mathematics and how they can be represented using coordinates or matrices. The magnitude of a vector is the length of its arrow, while the direction is the orientation of the vector in the reference axis.

Furthermore, we explored the dot product operation in detail. We discussed how the dot product is calculated using the magnitudes of the two vectors and the cosine of the angle between them.

We also showed that the dot product can be used to determine the work done by a force acting on an object and to find the projection of a vector onto another vector or plane. We discussed the difference between the dot( ) function and the vdot( ) function in the Numpy library.

The dot( ) function computes the dot product between two arrays and can be used for matrix multiplication. The vdot( ) function, on the other hand, computes the dot product of two N-dimensional arrays, including complex vectors, by flattening the arrays and then taking the complex conjugate of the second array.

We also showed how to calculate the vector dot product for n-dimensional arrays using an example with two-dimensional arrays. We discussed the importance of computing the dot product operation in the last dimension of the arrays.

Lastly, we looked in-depth at computing the vector dot product for complex numbers. We used an example of complex vectors and showed how the vdot( ) function utilizes the complex conjugate technique to compute the scalar value of the dot product correctly.

In conclusion, the vector dot product is a fundamental mathematical operation that has several applications in various fields. By understanding how to calculate the dot product and the functions available in the Numpy library, we can perform efficient computations of the dot product for n-dimensional arrays and complex vectors.

In summary, this article introduced the concept of vector dot product, which is a fundamental mathematical operation used to calculate the scalar product of two vectors. We discussed the difference between the two functions from the Numpy library – the dot( ) function and the vdot( ) function, used to perform the dot product of two vectors.

We also explored the calculation of the vector dot product for n-dimensional arrays and how to compute the dot product for complex numbers using the complex conjugate technique. The vector dot product is an essential tool in various fields like physics and engineering, and the article highlights its importance and how to use it efficiently.

Overall, understanding the vector dot product helps in accurate computation of important calculations that can be applied in real-life scenarios.