What is the dot product of two vectors?
The dot product of two vectors is a specific type of vector multiplication in which the result will be a scalar quantity. In this type of multiplication, two vectors are multiplied to get a scalar quantity (number) as a resultant.
In Physics and Mathematics, vector multiplication covers a major proportion of the understanding of the direction, magnitude, and resultant of two or more vectors. Like dot product, vectors can also be multiplied using another approach that is called a cross product of vectors.
To understand the dot product of two vectors, keep reading this blog till the end. We will discuss this topic comprehensively with the demonstration of an example.
Properties of the dot product
- The dot product of two vectors obeys the commutative law which means, A . B = B. A
- The dot product of a vector by itself is always “1”.
- The dot product of two parallel vectors will be the maximum.
- The dot product of two perpendicular vectors will be “0”.
- If a unit vector is multiplied by any other vector, the dot product resultant will be zero.
- When a unit vector is multiplied by itself, the answer will be equal to “1”.
- The dot product of two unit vectors will be the multiplication of its coordinates.
- If any of the vector’s magnitude is “0”, the resultant of the dot product will also be “0”.
How to Find the dot product of two vectors?
You can easily find the dot product of two vectors using the following formula.
A . B = AB Cos 𝜽
It means that we have to multiply the magnitudes of the vectors and find the “Cosine” of the angle between them to get the answer of the dot product. In expanded form, we can write the above formula in coordinates format as,
A . B = [(x1 . y1) + (x2 . y2)] Cos 𝜽
Here, the subscripts of “x” and “y” are the coordinates of the vectors. We can also extend the formula for the third dimension in a similar way. If you don’t understand the solution, you can get help from the Dot Product Calculator.
This online maths calculator has a simple interface that can be understood by everyone. You only have to insert the values of the coordinates in the given sections to get the answer. It will perform the calculation automatically after the insertion of the values and show the final answer.
How to calculate the dot product of two vectors?
To make the concept of the dot product of two vectors clear, let us show you an example here.
example1:
Find the dot product between two vectors if their coordinates are (2, 7) and (5, 8) while the angle between them is 60 degrees.
Solution:
To find the dot product, we only have to put the values in the following formula:
A . B = [(x1 . y1) + (x2 . y2)] Cos 𝜽
= [(2 x 5) + (7 x 8)] Cos (60)
As,
Cos 60 = 1/2
So,
= [10 + 56] x 1/2
= 66/2
= 33
Conclusion
In the above blog, the dot product of two vectors has been explained in detail. We have solved an example for your better understanding. We hope that you have understood the way to find the dot product of 2 vectors by reading this blog.
If you are facing issues in solving the questions related to it, you can use the Dot Product Calculator. It will enable you to get the solution instantly after inserting the values of the coordinates of the vector.
FAQ
What does the dot product represent?
It represents the similarity in the direction of two vectors when they are drawn on the paper.
What does the dot product of two vectors depend on?
The dot product of two vectors depends on the magnitude of the vectors and the angle between them.
Do vectors have to be the same size for the dot product?
No, it is not compulsory. Any two vectors can be multiplied using the dot product of vector methods.
Can a dot product return a vector?
No, the dot product will result in a scalar quantity instead of a vector. That’s why it is also called a Scalar Product.
Does dot product always return a scalar value?
Yes, the dot product will always return a scalar quantity (number).
