Dot Product
Dot product generally applies to vectors, and there are two ways of multiplication of vectors. The dot product of vectors and the cross product of vectors are the two ways of multiplying the vectors. These two methods of multiplication of vectors are considered since the vectors have two components, the magnitude and the direction. The dot product of two vectors is also called a scalar product and its answer is a scalar value.
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. Here we shall try to understand about the magnitude of the vector, angle between the vectors, projection of a vector, and then move ahead to check the formulas for dot product of two vectors..
Dot Product of Two Vectors
The dot product of two vectors is also called a scalar product as it results in a scalar value. The resultant of the dot product of two vectors lie in the same plane of the two vectors. The dot product of two vectors is simply the product of the magnitude of the two vectors. For two vectors that are inclined at an angle to each other, the dot product is equal to the product of the magnitude of the two vectors and the cosine of the angle between the vectors.
a.b = a.b.Cosθ
Terms Related To Dot Product
To understand the vector dot product, we first need to know how to find the magnitude of two vectors, the angle between two vectors, and to find the projection of one vector on another vector Here we shall try to understand each of these concepts, which would be helpful to understand and easily calculate the dot product of vectors.
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 \(a = a_1x + a_2y + a_3z\) the magnitude is a and is given by the below formula.
a = \(\sqrt{a_1^2 + a_2^2 +a_3^2}\)
Angle Between Two Vectors
The angle between two vector 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 product of the individual constituents of the two vectors, divided by the product of the magnitude of the two vectors. The formal for the angle between the two vectors is as follows.
\(Cos\theta = \dfrac{a.b}{a.b}\)
\(Cos\theta = \dfrac{a_1.b_1 + a_2.b_2 +a_3.b_3}{\sqrt{a_1^2 + a_2^2 +a_3^3}.\sqrt{b_1^2 + b_2^2 + b_3^2}}\)
Projection of a Vector
The vector project gives the projection of one vector over another vector. The vector projection of one vector over another vector is the length of the shadow of the given vector over another vector. The vector projection of one vector over another vector is obtained by multiplying the magnitude of the given vector with the cosecant of the angle between the two vectors. The resultant of a vector projection formula is a scalar value.
The vector projection of vector a over vector b = a.Cosθ
Formula For Dot Product
The dot product of two vectors can be found by two methods, and the result of this dot product is a scalar value. Algebraically the dot product of two vectors a and b is equal to the sum of the product of their individual constituents. Geometrically the dot product of the two vectors a and b is equal to the product of the magnitude of the vectors and the cosine of the angle between the two vectors. For two vectors a = \(a_1x + a_2y + a_3z\) and b = \(b_1x + b_2y + b_3z\), the two formulas for finding the vector dot product are as follows.
a.b = \(a_1.b_1 + a_2.b_2 +a_3. b_3\)
a.b = a.b.Cosθ
Further, we can also use the dot product calculator, to directly find the product of two vectors
Dot Product of Unit Vectors
The dot product of the unit vector is studied by taking the unit vectors i along the xaxis, j along the yaxis, and k along the zaxis respectively. The dot product of unit vectors i, j, 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.
i.i = j.j = k.k = 1
i.j = j.k = k.i = 0
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Solved Examples on Dot Product

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 a = 6, b = 7, and the angle between the vectors is θ = 60ª
The dot product of the two vectors is:
a.b = a.b.Cosθ
= (6).(7).Cos60°
= (6).(7).(1/2)
= (3).(7)
= 21
Answer: a.b = 21

Example 2: Find the angle between the two vectors 2i + 3i + k, and 5i 2j + 3k.
Solution:
The two given vectors are:
a = 2i + 3i + k, and b = 5i 2j + 3k
a = \(\sqrt{2^2 + 3^2 + 1^2}\) = \(\sqrt{4 + 9 + 1}\) = \(\sqrt{14}\)
b = \(\sqrt{5^2 + (2)^2 + 3^2}\) = \(\sqrt{25 + 4 + 9}\) = \(\sqrt{38}\)
a.b = 2.(5) + 3.(2) + 1.(3) = 10  6 + 3 = 7
Cosθ = \( \dfrac{a.b}{a.b}\)
= \(\dfrac{7}{\sqrt{14}.\sqrt{38}}\)
= \(\dfrac{7}{2.\sqrt{7 \times 19}}\)
= \(\dfrac{\sqrt{133}}{38}\)
θ = Cos^{1}\(\dfrac{\sqrt{133}}{38}\)
Answer: Therefore the angle between the vectors is Cos^{1}\(\dfrac{\sqrt{133}}{38}\)
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.(a.b = a1.b1 + a2.b2 + a3.b3. 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. (a.b = a.b.Cosθ). 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 a.b = a.b.Cos0° = a.b.1 = a.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 a.b = a.bCosθ and the cross product of two vectors is equal to a × b = a.b Sinθ. The resultant of the dot product of two vectors lies in the same plane as the two vectors, and the resultant of the cross product lies in a plane perpendicular to the plane containing the two vectors.
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, ie a and b. Secondly, find the cosecant of the angle θ between the two vectors. Finally take a product of the magnitude of the two vectors and the and the cosecant of the angle between the two vectors, to obtain the dot product of the two vectors. (a.b = a.b.Cosθ. Also check to dot product calculator, to easily find the vector dot product.
Why Is the Dot Product Is Called Scalar Product?
The dot product is a scalar because all the individual constituents of the answer are scalar values. In a.b = a.b.Cosθ, a, 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. a.b = a.b.cos90° = a.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 cosecant of the angle between the two vectors. And all the individual components of magnitude and angle are scalar quantities. Hence a.b = b.a, and the dot product follow 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.