Scalar Product
Scalar Product is a multiplication operation on vectors. The scalar product of two vectors is the sum of the product of the corresponding components of the vectors. In other words, the scalar product is equal to the product of the magnitudes of the two vectors and the cosine of the angle between them. It is a scalar quantity and is also called the dot product of vectors.
Let us explore the concept of scalar product, its formula for two and three vectors, properties and geometrical interpretation of scalar product and some solved examples to understand the concept of the scalar product better.
What is Scalar Product?
The scalar product is the multiplication of corresponding components of two or more vectors. As the name suggests, a scalar product gives a scalar quantity, that is, a real number as a result. The scalar product can also be calculated by taking the product of the magnitude of the vectors and the cosine of the angle between them. Now, let us see the scalar product formula for two and three vectors.
Scalar Product of Two Vectors Formula
The scalar product of two vectors is given by the product of the modulus of the first vector, the modulus of the second vector, and the cosine of the angle between them. In other words, the scalar product is the product of the magnitude of the first vector and the projection of the first vector onto the second vector. The scalar product formula for two vectors a and b is:
a.b = a b cosθ
Algebraic Formula for Scalar Product
Now, that we have discovered the scalar product formula for two vectors in terms of their magnitudes, let us see the algebraic formula for the scalar product of two vectors. If \(\overrightarrow{a} = [a_1, a_2, a_3, ..., a_n]\) and \(\overrightarrow{b} = [b_1, b_2, b_3, ..., b_n]\) are two vectors with n components, the scalar product of vectors a and b is given by:
\(\overrightarrow{a}.\overrightarrow{b}=\sum_{i=1}^{n}a_ib_i=a_1b_1+a_2b_2+a_3b_3+...+a_nb_n\)
Geometrical Interpretation of Scalar Product
Since we know the formula and meaning of the scalar product, let us understand the geometrical interpretation of the scalar product of two vectors. Consider the figure given below:
In the above figure, OA represents the length of vector a, OA' represents the length of vector b and θ is the angle between them. As we can see in the figure, OA' is the projection of vector a onto vector b. We know that in a rightangled triangle, the cosine of the angle in consideration is equal to the ratio of the base and hypotenuse. So, we have:
cos θ = OA'/OA = OA'/a ⇒ OA' = a.cos θ
From the scalar product formula, we have a.b = a b cos θ = a \(\text{proj}_{\overrightarrow{b}}(\overrightarrow{a})\), that is, the scalar product of vectors a and b is equal to the magnitude of vector a times the projection of a onto vector b.
Applications of Scalar Product
Scalar Product has many applications in vector theory, some of which include:
 Projection of a Vector: Scalar product is used to determine the projection of a vector onto another vector. The projection of a vector a onto vector b is given by a.b/b. Similarly, the projection of vector b onto vector a is given by a.b/a.
 Scalar Triple Product: Scalar product is used in the calculation of scalar triple product of three vectors. The formula for the scalar triple product is a.(b × c) = b.(c × a) = c.(a × b)
 Angle Between Two Vectors: Scalar product is used to determine the angle between two vectors using the formula cos θ = (a.b)/(a b).
Scalar Product Properties
Now, that we have understood the concept of scalar product, let us go through some of the important properties of the scalar product of vectors a and b that will help us in solving various problems:
 Commutative Property  Scalar product is commutative, that is, a.b = b.a

Distributive Property  Scalar product of vectors follow the distributive property:
a.(b + c) = a.b + a.c
(a + b).c = a.c + b.c
a.(b  c) = a.b  a.c
(a  b).c = a.c  b.c 
Scalar Multiplication  Scalar product satisfies the scalar multiplication given as: \((r_1\overrightarrow{a}).(r_2\overrightarrow{b}) = r_1r_2(\overrightarrow{a}.\overrightarrow{b})\)

Two vectors are said to orthogonal if their scalar product is zero, that is, vectors a and b are orthogonal if a.b = 0
Tips and Tricks on Scalar Product
 \(\hat i.\hat i\) = \(\hat j.\hat j\) = \(\hat k.\hat k\)= 1
 \(\hat i.\hat j\) = \(\hat j.\hat k\) = \(\hat k.\hat i\)= 0
 If θ = 0 then a.b = ab [Two vectors are parallel in the same direction ⇒ θ = 0 ] .
 If θ = π , a.b = ab [Two vectors are parallel in the opposite direction ⇒ θ = π.].
 If θ = π/2, then a.b = 0 [Two vectors are⇒ θ = π/2], that is, they are orthogonal.
Related Topics
Scalar Product Examples

Example 1: Find the scalar product of the vectors a = 2i + 3j  6k and b = i + 9k.
Solution: To find the scalar product of the given vectors a and b, we will multiply their corresponding components.
a.b = (2i + 3j  6k).(i + 0j + 9k)
= 2.1 + 3.0 + (6).9
= 2 + 3  54
= 49
Answer: The scalar product of vectors a = 2i + 3j  6k and b = i + 9k is 49.

Example 2: Calculate the scalar product of vectors a and b when the modulus of a is 9, modulus of b is 7 and the angle between the two vectors is 60°.
Solution: To determine the scalar product of vectors a and b, we will use the scalar product formula.
a.b = a b cosθ
= 9 × 7 cos 60°
= 63 × 1/2
= 31.5
Answer: Hence, the scalar product of vectors a and b is 31.5
FAQs on Scalar Product
What is Scalar Product in Math?
Scalar Product is a multiplication operation on vectors. Scalar product of vectors is given by a.b = a b cosθ.
What is the Scalar Product of Two Vectors?
The scalar product of two vectors is the sum of the product of the corresponding components of the vectors. \(\overrightarrow{a}.\overrightarrow{b}=\sum_{i=1}^{n}a_ib_i=a_1b_1+a_2b_2+a_3b_3+...+a_nb_n\)
How to Find Angle with Scalar Product?
Scalar product is used to determine the angle between two vectors using the formula cos θ = (a.b)/(a b) ⇒ θ = cos^{1}[(a.b)/(a b)]
What is the Difference Between Scalar Product And Vector Product?
The scalar product is the product of vectors which gives a scalar quantity whereas the vector product is the product of vectors which gives a vector quantity as the product.
Is the Scalar Product the Same as the Dot Product?
Yes, the scalar product is the same as the dot product.
Is Scalar Product Commutative?
Yes, the scalar product is commutative: a.b = b.a
What is Scalar Product Formula?
Scalar product formula os given by, a.b = a b cosθ or \(\overrightarrow{a}.\overrightarrow{b}=\sum_{i=1}^{n}a_ib_i=a_1b_1+a_2b_2+a_3b_3+...+a_nb_n\)
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