Dot product of two vectors in C++

What is the dot product?

The dot product, also known as the inner product or scalar product, is a mathematical operation in which two vectors result in a scalar (a single numeric value). The dot product is defined for vectors in both 2D and 3D to understand and deal with vector behaviors. It is used in various fields, including physics, mathematics, and computer graphics.

Let's discuss the formulas to calculate the dot product of two vectors.

Algebraic formula to calculate dot product of two vectors

To calculate the dot product between two vectors, we multiply their corresponding values along the x-, y-, and z- axes before adding them. Suppose we have two vectors: $v1$ and $v2$ such that:

$v1 = (a1, a2,a3)$

$v2 = (b1,b2,b3)$

So the dot product is calculated as:

Here, $k$ is the resultant scalar of the dot product.

Geometric formula

We use the following formula to calculate the dot product for the vectors geometrically when we know the angle between them.

• $k$ : The resultant scalar of the dot product.

• $|v1|$ : The magnitude of the vector $v1$.

• $|v2|$: The magnitude of the vector $v2$.

• $θ$: The angle between vectors $v1$ and $v2$.

Calculation of the dot product in C++

Let's calculate the dot product of two vectors in C++ now. The following examples refer to the algebraic formula while calculating the dot product.

1. Using the inner_product function

We can calculate the dot product of two vectors using the inner_product function from the <numeric> header file, which is part of the Standard Template Library (STL). To include the <numeric> header file, we write the following code:

Now let's discuss the syntax for using inner_product function:

• class InputIt1, class InputIt2, class T: The template parameters that represent the placeholders for the types of arguments that the function template can accept.

• first1: The iterator that points to the beginning of the first vector.

• last1: The iterator that points to the end of the first vector.

• first2: The iterator that points to the beginning of the second vector.

• init: The initial value for the accumulation of the dot product. We start adding the dot product from this value.

Implementing the dot product in C++

The following code demonstrates how to calculate the dot product of two vectors using the inner_product function.

• Lines 1–3: We include the necessary libraries.

• Line 4: We bring all the names from the std namespace into the code so we can avoid prefixing them with std::.

• Lines 7–8: We define the two vectors whose dot product we want to calculate of.

• Line 10: We calculate the dot product of the two vectors using inner_product function. We send iterators (v1.begin() and v2.begin()) that point to the beginning of each vector, and v1.end() that points just past the last element of v1 as parameters. The last argument 0 is the initial value for the accumulation of the dot product.

• Line 12: We print the dot product.

2. Using the iterative approach

We use an iterative approach to calculate the dot product of the vectors. We use a loop to iterate through the axes values of both vectors, multiply the corresponding values, and add the results together.

Implementing the dot product using the iterative approach

The following code demonstrates how to calculate the dot product of two vectors using an iteration.

• Lines 5–11: In the function dotProduct, we multiply the corresponding values of each axis of the vectors and add them in a variable result.

• Lines 15–16: We define the two vectors we want to calculate the dot product of.

• Line 18: We make a call to the function dotProduct.

Conclusion

By understanding the concept and implementation of the dot product in C++, we can perform various mathematical computations efficiently and use the dot product in a wide range of applications, including vector manipulations, physics simulations, and computer graphics rendering.