Dot Product Formula
The dot product is also known as the scalar product. It’s called a scalar product because this multiplication gives magnitude as a result. It's a way of representing the multiplication between two (or more) vectors. Dot product formula thus helps in calculating the scalar product of two vectors. The dot product formula represents the dot product of two vectors as a multiplication of the two vectors, and the cosine of the angle formed between them.
What Is Dot Product Formula?
Dot Product Formula of the given vectors can be expressed as follows. Here a and b are the two vectors, a and b are their respective magnitudes, and θ is the angle between the two vectors.
a.b=ab cosθ
a.b =a_{1}b_{1 }+ a_{2}b_{2 }+ a_{3}b_{3 }+....+ a_{n}b_{n }
Learn the dot product formula using solved examples in the sections below.
Solved Examples on Dot Product Formula

Example 1: Find the dot product of two vectors \(\vec F_1= \vec i+5 \vec j+ \vec k \) and \(\vec F_2= 3\vec i+ \vec j+ 2 \vec k\)
.Solution:
To find: Dot product of two vectors
Given,
\({\vec F_1= \vec i+5 \vec j+ \vec k\)
\({\vec F_2= 3\vec i+ \vec j+ 2 \vec k\)Using Dot Product Formula,
a.b =a_{1}b_{1 }+ a_{2}b_{2 }+ a_{3}b_{3 }+⋯+ a_{n}b_{n}
F_{1}.F_{2 }= a_{1}b_{1 }+ a_{2}b_{2 }+ a_{3}b_{3 }+⋯+ a_{n}b_{n}
= 1×3 + 5×1 + 1×2
= 3 + 5 + 2
= 10Answer:\(\vec F_1.\vec F_2 = 10 \).

Example 2: A two force system A and B originate from a single point have a magnitude equals to 20N and 50N respectively. The angle between the two vectors is 60°. Find the dot product of the two vectors.
Solution:
To find: Dot product of the vectors A and B.
Given:
A = 20N
B = 50N
θ = 60ºUsing Dot Product Formula,
a.b = ab cosθ
= 20*50* cos60
= 1000 * 0.5
= 500NAnswer: \({\vec {A}}.{\vec{B}} \) = 500N.