Ever wondered what is dot product used for? Or, when and how to use dot product in real life? If yes, then, you are at the right place at the very right time. Dot product, also known as scalar product, is a mathematical operation that takes two vectors and returns a scalar quantity.
The dot product has numerous applications in various fields, including physics, engineering, computer science, and more. In this article, we will explore some of the exclusive applications of the dot product. So, without wasting any more time, let’s dive right in…!!!
Applications of Dot Product in Real Life
- Testing the Orthogonality
- Image and Signal Processing
- Solving Systems of Linear Equations
- Calculating Work Done by a Force
- Testing for Collinearity
- Finding Projections
Testing the Orthogonality
The very first one in my list of top 6 applications of the dot product is that you can test the orthogonality using dot products. According to the definition of dot products, two vectors are orthogonal, only, and, only if, their dot product is zero.
You can use this property in many applications such as finding the normal vector to a plane, testing for perpendicularity in geometry, etc.
Image and Signal Processing
Yup, it’s true. You can extensively use dot products for image and signal processing. The dot product is used in image compression algorithms to reduce the amount of data and size to represent the image.
Not to mention, it is also used in audio processing to determine the similarity between two audio signals. For example, in this article, all the images that I am using are compressed by using dot products. Well, of course, I personally have not used it to compress the size of an image, the tools I use did.
Solving Systems of Linear Equations
You can use the dot product to find the unknown vector in the system of linear equations. A system of linear equations can be represented in a matrix form i.e
Ax = b
A = matrix of coefficients
x = vector of variables
b = vector of constants
Here, you can use the dot product to find the dot product of each row of (A) with the vector (x). Equating these dot products with the corresponding elements of (b) will give you a set of equations that can be easily solved to find the values of variables.
Calculating Work Done by a Force
One of the most important application of dot product in physics is in calculating work done by a force. You can use the dot product of force and displacement to calculate the work done by a force. The formula for work done by a force is:
W = F . d
W = Work done
F = Force vector
d = displacement vector
Testing for Collinearity
As per the definition, two vectors are collinear, only, and, only if, one vector is a scalar multiple of the other. So, how can you test if this theory is correct or not?
You can use the dot product of two vectors to check whether two vectors are collinear or not. If the dot product of two vectors is equal to the product of their magnitudes, then the vectors are collinear.
Last but not least one in my list of top 6 dot products applications is that with the help of dot products, you can easily find the projection of one vector onto another vector.
Mathematically, the projection of a vector (U) on a vector (V) is given by:
proj v(u) = ((uv) / (|v|^2))v
Some Other Uses of Dot Product in Real Life
Apart from the above-mentioned ones, I am also mentioning a few here.
- Determining the magnitude of a vector
- Calculating the eigenvalues and eigenvectors of a matrix in linear algebra
- Finding the intersection of two lines in geometry
- Determining the direction of maximum change in a function
- Determining the velocity and acceleration of an object in kinematics, etc.
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