Dot Product
The dot product is one way of multiplying two or more vectors. The resultant of the dot product of vectors is a scalar quantity. Thus, the dot product is also known as a scalar product. Algebraically, it is the sum of the products of the corresponding entries of two sequences of numbers. Geometrically, it is the product of the Euclidean magnitude of two vectors and the cosine of the angle between them. The dot product of vectors finds various applications in geometry, mechanics, engineering, and astronomy. Let us discuss the dot product in detail in the upcoming sections.
What is Dot Product?
Dot Product of vectors is equal to the product of the magnitudes of the two vectors, and the cosine of the angle between the two vectors. The resultant of the dot product of two vectors lie in the same plane of the two vectors. The dot product may be a positive real number or a negative real number or a zero.
Dot Product Definition
In vector algebra, if two vectors are given as: a= [\(a_1\),\(a_2\),\(a_3\),\(a_4\),….,\(a_n\)] and b = [\(b_1\),\(b_2\),\(b_3\),\(b_4\),….,\(b_n\)]
then their dot product is given by:
\(\overrightarrow a. \overrightarrow b\) = \(a_1 b_1\)+\(a_2 b_2\)+\(a_3 b_3\)+……….+\(a_n b_n\)
\(\overrightarrow a. \overrightarrow b = \sum_{i=1}^{n} a_i b_i\)
Dot Product Formula for Vectors
Let a and b be two nonzero vectors, and θ be the included angle of the vectors. Then the scalar product or dot product is denoted by \(\overrightarrow a. \overrightarrow b\), which is defined as:
\(\overrightarrow a. \overrightarrow b\) = \(\overrightarrow a\overrightarrow b\) cos θ.
Here,
 \(\overrightarrow a\) is the magnitude of \(\overrightarrow a\),
 \(\overrightarrow b\) is the magnitude of \(\overrightarrow b\), and
 θ is the angle between the vectors.
Note: θ is not defined if either \(\overrightarrow a\) = 0 or \(\overrightarrow b\) = 0.
Geometrical Meaning of Dot Product
The dot product of two vectors is constructed by taking the component of one vector in the direction of the other and multiplying it with the magnitude of the other vector. To understand the vector dot product, we first need to know how to find the magnitude of two vectors, and the angle between two vectors to find the projection of one vector over another vector.
Magnitude of A Vector
A vector represents a direction and a magnitude. The magnitude of a vector is the square root of the sum of the squares of the individual constituents of the vector. The magnitude of a vector is a positive quantity. For a vector \(\overrightarrow a = a_1\hat i + a_2 \hat j + a_3 \hat k\), the magnitude is \(\overrightarrow a\) and is given by the formula, \(\overrightarrow a = \sqrt{a_1^2 + a_2^2 +a_3^2}\)
Projection of a Vector
The dot product is useful for finding the component of one vector in the direction of the other. The vector projection of one vector over another vector is the length of the shadow of the given vector over another vector. It is obtained by multiplying the magnitude of the given vectors with the cosine of the angle between the two vectors. The resultant of a vector projection formula is a scalar value.
Let OA = \(\overrightarrow a\), OB = \(\overrightarrow b\), be the two vectors and θ be the angle between \(\overrightarrow a\) and \(\overrightarrow b\). Draw AL perpendicular to OB.
From the right triangle OAL , cos θ = OL/OA
OL = OA cos θ = \(\overrightarrow a\) cos θ
OL is the vector projection of a on b.
\(\overrightarrow a. \overrightarrow b\) = \(\overrightarrow a\overrightarrow b\) cos θ = \(\overrightarrow b\) OL
= \(\overrightarrow b\) (projection of \(\overrightarrow a\) on \(\overrightarrow b\))
Thus, projection of \(\overrightarrow a\) on \(\overrightarrow b = \dfrac{\overrightarrow a. \overrightarrow b}{\overrightarrow b}\)
Similarly, the vector projection of \(\overrightarrow b\) on \(\overrightarrow a = \dfrac{\overrightarrow a. \overrightarrow b}{\overrightarrow a}\)
Angle Between Two Vectors Using Dot Product
The angle between two vectors is calculated as the cosine of the angle between the two vectors. The cosine of the angle between two vectors is equal to the sum of the products of the individual constituents of the two vectors, divided by the product of the magnitude of the two vectors. The formula for the angle between the two vectors is as follows.
\(cos\theta = \dfrac{\overrightarrow a.\overrightarrow b}{\overrightarrow a.\overrightarrow b}\)
If \(\overrightarrow a = a_1\hat i + a_2 \hat j + a_3 \hat k\) and \(\overrightarrow b = b_1 \hat i + b_2 \hat j + b_3\hat k\) then
\(cos\theta = \dfrac{a_1.b_1 + a_2.b_2 +a_3.b_3}{\sqrt{a_1^2 + a_2^2 +a_3^3}.\sqrt{b_1^2 + b_2^2 + b_3^2}}\)
Working Rule to Find The Dot Product of Two Vectors
If the two vectors are expressed in terms of unit vectors, i, j, k, along the x, y, z axes, then the scalar product is obtained as follows:
If \(\overrightarrow a = a_1\hat i + a_2 \hat j + a_3 \hat k\) and \(\overrightarrow b = b_1 \hat i + b_2 \hat j + b_3\hat k\), then
\(\overrightarrow a. \overrightarrow b\) = \((a_1 \hat i + a_2 \hat j + a_3 \hat k)(b_1 \hat i + b_2 \hat j + b_3 \hat k)\)
= \((a_1b_1) (\hat i. \hat i) + (a_1b_2) (\hat i.\hat j)+ (a_1b_3) (\hat i. \hat k) + \\(a_2b_1) (\hat j. \hat i) + (a_2b_2)(\hat j. \hat j) + (a_2b_3 (\hat j. \hat k) + \\(a_3b_1)(\hat k. \hat i) + (a_3b_2)(\hat k. \hat j) + (a_3b_3)(\hat k. \hat k)\)
= \(a_1b_1\) + \(a_2b_2\)+ \(a_3b_3\)
Matrix Representation of Dot Product
It is easy to compute the dot product of vectors if the vectors are represented as row or column matrices. The transpose matrix of the first vector is obtained as a row matrix. Matrix multiplication is done. The row matrix and column matrix are multiplied to get the sum of the product of the corresponding components of the two vectors.
Properties of Dot Product
The following are the properties of the dot product of vectors.
 Commutative property
 Distributive property
 Natural property
 General properties
 Vector identities
Commutative property of Dot Product:
With the usual definition, \(\overrightarrow a\). \(\overrightarrow b\) = \(\overrightarrow b\) . \(\overrightarrow a\) , as we have \(\overrightarrow a\overrightarrow b\) cos θ = \(\overrightarrow b\overrightarrow a\) cos θ
Distributivity of Dot Product
Let a, b, and c be any three vectors, then the scalar product is distributive over addition and subtraction. This property can be extended to any number of vectors.
 \(\overrightarrow a. (\overrightarrow b+\overrightarrow c) = \overrightarrow a. \overrightarrow b + \overrightarrow a. \overrightarrow c\)
 \((\overrightarrow a+\overrightarrow b). \overrightarrow c = \overrightarrow a. \overrightarrow c+ \overrightarrow b. \overrightarrow c\)
 \(\overrightarrow a. (\overrightarrow b  \overrightarrow c) = \overrightarrow a. \overrightarrow b  \overrightarrow a. \overrightarrow c\)
 \((\overrightarrow a \overrightarrow b). \overrightarrow c = \overrightarrow a. \overrightarrow c  \overrightarrow b. \overrightarrow c\)
Nature of Dot Product
 We know that 0 ≤ θ ≤ π.
 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 perpendicular ⇒ θ = π/2]
 If 0 < θ < π/2, then cosθ is positive and hence a.b is positive.
 If π/2 < θ < π then cosθ is negative and hence a.b is negative.
Other Properties of Dot Product
 Let a and b be any two vectors, and λ be any scalar. Then (λ\(\overrightarrow a) . \overrightarrow b\) = λ (\(\overrightarrow a . \overrightarrow b)\)
 For any two scalars λ and μ, λ\(\overrightarrow a\) . μ \(\overrightarrow b\) = (λμ\(\overrightarrow a). \overrightarrow b\) = \(\overrightarrow a\). (λμ \(\overrightarrow b\))
 The length of a vector is the square root of the dot product of the vector by itself. \(\overrightarrow a\) = \(\sqrt{\overrightarrow a . \overrightarrow a}\)
 \(\overrightarrow a. \overrightarrow a\) = \(\overrightarrow a\)^{2 }(or) it can be written as a^{2}
 For any two vectors a and b, \(\overrightarrow a + \overrightarrow b\) ≤ \(\overrightarrow a\) + \(\overrightarrow b\)
Vector Identities
 (\(\overrightarrow a + \overrightarrow b\)) ^{2 }= \(\overrightarrow a\)^{2 }+ \(\overrightarrow b\)^{2 }+ 2 \((\overrightarrow a.\overrightarrow b)\)
 (\(\overrightarrow a  \overrightarrow b\)) ^{2 }= \(\overrightarrow a\)^{2 }+ \(\overrightarrow b\)^{2 } 2 \((\overrightarrow a.\overrightarrow b)\)
 \((\overrightarrow a + \overrightarrow b). (\overrightarrow a  \overrightarrow b) = \overrightarrow a\)^{2 } \(\overrightarrow b\)^{2 }≤ \(\overrightarrow a\) + \(\overrightarrow b\)
Dot Product of Unit Vectors
The dot product of the unit vectors is studied by taking the unit vectors \(\hat i\) along the xaxis, \(\hat j\) along the yaxis, and \(\hat k\) along the zaxis respectively. The dot product of unit vectors \(\hat i\), \(\hat j\), \(\hat k\) follows similar rules as the dot product of vectors. The angle between the same vectors is equal to 0º, and hence their dot product is equal to 1. And the angle between two perpendicular vectors is 90º, and their dot product is equal to 0.
\(\hat i.\hat i\) = \(\hat j.\hat j\) = \(\hat k.\hat k\)= 1
\(\hat i.\hat j\) = \(\hat j.\hat k\) = \(\hat k.\hat i\)= 0
Applications of Dot Product
The application of the scalar product is the calculation of work. The product of the force applied and the displacement is called the work. If force is exerted at an angle θ to the displacement, the work done is given as the dot product of force and displacement as W = f d cos θ. The dot product is also used to test if two vectors are orthogonal or not. \(\overrightarrow a. \overrightarrow b\) = \(\overrightarrow a\overrightarrow b\) cos 90º ⇒ \(\overrightarrow a. \overrightarrow b\) = 0
Important Notes on Dot Product:
 The dot product or the scalar product is a way to multiply two vectors.
 Geometrically, the dot product is the product of the length of the vectors with the cosine angle between them. \(\overrightarrow a. \overrightarrow b\) = \(\overrightarrow a\overrightarrow b\) cos θ
 It is a scalar quantity having no direction. It is easily computed from the sum of the product of the components of the two vectors.
 If \(\overrightarrow a\) = \(a_1\) i + \(a_2\) j + \(a_3\) k and \(\overrightarrow b\)= \(b_1\) i + \(b_2\) j + \(b_3\) k, then \(\overrightarrow a. \overrightarrow b = a_1b_1 + a_2b_2+ a_3b_3\)
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Dot Product of Vectors Examples

Example 1: Find the dot product of two vectors having magnitudes of 6 units and 7 units, and the angle between the vectors is 60°.
Solution:
The magnitudes of the two vectors are \(\overrightarrow a\) = 6, \(\overrightarrow b\) = 7, and the angle between the vectors is θ = 60°
The dot product of the two vectors is:
\(\overrightarrow a. \overrightarrow b\) = \(\overrightarrow a.\overrightarrow b\).Cosθ
= (6).(7).cos60°
= (6).(7).(1/2)
= (3).(7)
= 21
Answer: \(\overrightarrow a. \overrightarrow b\) = 21

Example 2. Find the angle between the two vectors 2i + 3i + k, and 5i 2j + 3k.
Solution:
The two given vectors are:
\(\overrightarrow a\) = 2i + 3i + k, and \(\overrightarrow b\) = 5i 2j + 3k
\(\overrightarrow a\) = \(\sqrt{2^2 + 3^2 + 1^2}\) = \(\sqrt{4 + 9 + 1}\) = \(\sqrt{14}\)
\(\overrightarrow b\) = \(\sqrt{5^2 + (2)^2 + 3^2}\) = \(\sqrt{25 + 4 + 9}\) = \(\sqrt{38}\)
Using the dot product we have \(\overrightarrow a.\overrightarrow b\) = 2.(5) + 3.(2) + 1.(3) = 10  6 + 3 = 7
Cosθ = \( \dfrac{\overrightarrow a. \overrightarrow b}{\overrightarrow a.\overrightarrow b}\)
= \(\dfrac{7}{\sqrt{14}.\sqrt{38}}\)
= \(\dfrac{7}{2.\sqrt{7 \times 19}}\)
= \(\dfrac{7}{2 \sqrt {133}}\)
θ = Cos^{1}\(\dfrac{7}{2 \sqrt {133}}\)
θ = Cos^{1 }0.304 = 72.3°
Answer: Therefore the angle between the vectors is 72.3°

Example 3. Test whether vectors \(\overrightarrow a\) = 3i + 2j k and \(\overrightarrow b\) = i  2j  k are orthogonal.
Solution:
To check if two vectors are perpendicular, we compute the dot product and see if the result is 0.
Given: \(\overrightarrow a\) = 3i + 2j k and \(\overrightarrow b\) = i  2j  k.
The dot product is computed as \(\overrightarrow a.\overrightarrow b = a_1b_1 + a_2b_2+ a_3b_3\)
= 3(1) + 2(2) +(1)(1)
= 3  4 +1
= 0
Answer: The dot product proves that the given vectors are orthogonal.
FAQs on Dot Product
What is the Dot Product of Two Vectors?
The dot product of two vectors has two definitions. Algebraically the dot product of two vectors is equal to the sum of the products of the individual components of the two vectors. \(\overrightarrow a. \overrightarrow b\) = \(a_1b_1\) + \(a_2b_2\)+ \(a_3b_3\). Geometrically the dot product of two vectors is the product of the magnitude of the vectors and the cosine of the angle between the two vectors. ( \(\overrightarrow a. \overrightarrow b\) = \(\overrightarrow a\overrightarrow b\) cos θ). The resultant of the dot product of vectors is a scalar value.
What is the Dot Product of Two Parallel Vectors?
The dot product of two parallel vectors is equal to the product of the magnitude of the two vectors. For two parallel vectors, the angle between the vectors is 0°, and Cos0°= 1. Hence for two parallel vectors a and b we have \(\overrightarrow a. \overrightarrow b\) = \(\overrightarrow a\overrightarrow b\) cos 0° = \(\overrightarrow a.\overrightarrow b\).1 = \(\overrightarrow a.\overrightarrow b\).
What is the Difference Between Dot Product and Cross Product?
The dot product is a scalar product and the cross product is the vector product. The dot product of two vectors is \(\overrightarrow a. \overrightarrow b\) = \(\overrightarrow a.\overrightarrow b\)Cosθ and the cross product of two vectors is equal to \(\overrightarrow a\) × \(\overrightarrow b\) = \(\overrightarrow a.\overrightarrow b\) Sinθ.\(\hat{n}\). The resultant of the dot product of two vectors lie in the same plane as the two vectors, whereas the resultant of the cross product lies in a plane perpendicular to the plane spanning the two vectors.
What is Dot Product Formula?
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. Dot product formula of the given vectors can be expressed as follows. Here a and b are the two vectors, \(\overrightarrow a\) and \(\overrightarrow b\) are their respective magnitudes, and θ is the angle between the two vectors a.b = \(\overrightarrow a\overrightarrow b\) cosθ.
Does Dot Product Formula Involve Multiplication?
Multiplication of two vectors is not the same as scalar multiplication. There are two types of multiplication involving two vectors. The scalar product is 'dot product' and the vector product is 'cross product'. 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 the Purpose of Dot Product Formula?
The purpose of the dot product is to tell us the amount of force vector is applied in the direction of the motion vector. The dot product also lets us measure the angle that is formed by a pair of vectors and the relative position of a vector against the coordinate axes.
What Happens When a Dot Product after Using Dot Product Formula Is 0?
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. If the dot product is 0, then we can conclude that either the length of one or both vectors is 0, or the angle between them is 90 degrees.
Where Do We Use Dot Product?
The concept of the dot product is used prominently in physics and engineering. For two quantities placed at an angle to each other, the dot product gives the result of these two vectors. Let us take an example of force applied on a body F, and the displacement of the body is d. If the angle between the force vector F and the displacement vector d is θ, then the work done is the product of force and displacement. W = F.d.Cosθ.
How To Calculate the Dot Product?
The dot product can be calculated in three simple steps. First find the magnitude of the two vectors a and b, i.e., \(\overrightarrow a\) and \(\overrightarrow b\). Secondly, find the cosine of the angle θ between the two vectors. Finally take a product of the magnitude of the two vectors and the and cosine of the angle between the two vectors, to obtain the dot product of the two vectors. (\(\overrightarrow a. \overrightarrow b\) = \(\overrightarrow a.\overrightarrow b\).Cosθ. Also check to dot product calculator, to easily find the vector dot product.
Why is the Dot Product Called Scalar Product?
The dot product is a scalar because all the individual constituents of the answer are scalar values. In \(\overrightarrow a. \overrightarrow b\) = \(\overrightarrow a.\overrightarrow b\).Cosθ, \(\overrightarrow a, \overrightarrow b\), and Cosθ are all scalar values. Hence the dot product is also called a scalar product.
Why Do We Use Cos in Dot Product?
For finding the dot product we need to have the two vectors a, b in the same direction. Since the vectors, a and b are at an angle to each other, the value acosθ is the component of vector a in the direction of vector b. Hence we can find cosθ in the dot product of two vectors.
Why is the Dot Product of Orthogonal Vectors Equal to 0?
The two orthogonal vectors are perpendicular to each other and the angle between the two vectors is equal to 90°. Since Cos90° = 0, the dot product of two orthogonal vectors is equal to 0. \(\overrightarrow a. \overrightarrow b\) = \(\overrightarrow a.\overrightarrow b\).cos90° = \(\overrightarrow a.\overrightarrow b\).0 = 0.
Why is the Dot Product Commutative?
The dot product of two vectors is equal to the product of the magnitude of the two vectors and the cosine of the angle between the two vectors. And all the individual components of magnitude and angle are scalar quantities. Hence \(\overrightarrow a. \overrightarrow b\) = \(\overrightarrow b. \overrightarrow a\), and the dot product of vectors follows the commutative property.
Can a Dot Product be equal to zero?
The dot product of two vectors can be zero if either of the two vectors is zero or if the two vectors are perpendicular to each other. For two nonzero vectors, the dot product is zero if the angle between the two vectors is 90º, because Cos90º = 0.
Is the Dot Product of Two Collinear Vectors 0?
No. This is because the angle between two collinear vectors is 0 and so, the dot product of two collinear vectors is just the product of the their magnitudes (as cos 0 = 1). In fact, the cross product of two collinear vectors is a zero vector.
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