Dot Product – Explanation, Formulas, and Examples
Dot Product of Vectors
The dot product is also called the scalar product in which we take any two vectors, say ‘m’ and ‘n’, and perform their dot product, we get our desired result, which is also a scalar quantity. There are two questions in the given statement, they are as follows:
- Is dot product easy to calculate just like we use an operator multiplication (*)?
- Do we get the scalar quantity as our resultant, or is there any other thing on which our result depends?
Solutions to all your queries are covered in the article along with illustrating examples on “Dot Product of Two or More Vectors” that will clear your basics on this topic.
How Do We Perform the Dot Product of Two Vectors?
Assume that John is teaching “Archery” to his daughter and each time she bows an arrow at one position and tries pointing to the target. Here, we have two arrows shooted by a man and his daughter. We notice that both the arrow has some length or magnitude and direction to which it points. So, when we have both the magnitude and the direction, we say that this arrow is a vector. These concepts can be learned in a better way by attending online math classes from the best math tutoring platform like Cuemath.
Thus, vectors are those quantities that have both magnitude and direction. Further, we can perform the product of the vectors of both father and daughter by using a Dot Product of two vectors method.
Man’s vector is ‘m ’ and daughter’s vector is ‘d .’ The dot product can be written as;
m . d = |m | x |d | x Cos Ө = md Cos Ө
Do you know why we used Cos Ө here? Well, Ө is the angle between the two vectors, but how is it possible to have an angle between two vectors? Well, if you look at the vectors of a man and his daughter, we get an inclination between the two. This is the reason we used angles here.
Examples of Dot Product of Two Vectors
Let us look at a few examples to understand this concept better.
Example 1: Assuming that two vectors ‘m’ and ‘d’ are of magnitude 11 and 15 units, respectively and an angle between them is 56.8°. Calculate the dot product of these two vectors.
Solution:
We have the following data:
|m | = 11
|d | = 15
Ө = 56.8°
Using the above formula for the Dot Product:
m . d = |m | x |d | x Cos Ө = md Cos Ө
Substituting the values in this equation:
= 11 x 15 x Cos (56.8°)
= 165 x Cos (56.8°)
Please note that Cos (56.8°) is 0.54756322.
Therefore, on solving we get the Dot Product of m and d as 90.3479313 (rounded).
Example 2: Assume that a girl and a boy planned a party, they both had three sets of dresses named ‘2i’, ‘4j’, and ‘6k’ while another had a set of ‘3i’, ‘4j’, and ‘5k’. However, by mistake, their dresses got mixed. So, what can be the Dot Product of this intermixing?
Solution: Let us write the above two scenarios in the following manner:
2i + 4j + 6k ….(1)
3i + 4j + 5k ….(2)
Now, let us perform the dot product of this situation:
(2i + 4j + 6k) * (3i + 4j + 5k)
Please note that in a dot product, the product of i with itself is ‘i’, j is ‘j’, and k is ‘k’. However, i with j or k will be zero, j with i or k will be zero, and so on. The same thing we will follow below:
= {(2i * 3i) + (2i * 4j) + (2i * 5k)} + {(4j * 3i) + (4j * 4j) + (4j * 5k)} + {(6k * 3i) + (6k * 4j) + (6k * 5k)}
= 6 + 0 + 0 + 0 + 16 + 0 + 0 + 0 + 30
= 52
Therefore, the intermixing of dresses gives a Dot Product of 52.
