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, the dot product of two vectors is the product of their Euclidean magnitudes 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.

The dot product of two vectors is equal to the product of the magnitudes of the two vectors, and the cosine of the angle between them. i.e., the dot product of two vectors \(\overrightarrow a\) and \(\overrightarrow b\) is denoted by \(\overrightarrow a \cdot \overrightarrow b\) and is defined as \(|\overrightarrow a||\overrightarrow b|\) cos θ. 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: \(\overrightarrow a \) = [\(a_1\),\(a_2\),\(a_3\),\(a_4\),….,\(a_n\)] and \(\overrightarrow b\) = [\(b_1\),\(b_2\),\(b_3\),\(b_4\),….,\(b_n\)]
then their dot product is given by:
\(\overrightarrow a \cdot \overrightarrow b\) = \(a_1 b_1\)+\(a_2 b_2\)+\(a_3 b_3\)+……….+\(a_n b_n\)

\(\overrightarrow a \cdot \overrightarrow b = \sum_{i=1}^{n} a_i b_i\)

Dot Product Formula for Vectors

Let \(\overrightarrow a\) and \(\overrightarrow b\) be two non-zero vectors, and θ be the included angle of the vectors. Then the scalar product of two vectors or dot product is denoted by \(\overrightarrow a \cdot \overrightarrow b\), which is defined as:

\(\overrightarrow a \cdot \overrightarrow b\) = \(|\overrightarrow a||\overrightarrow b|\) 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.

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 \(\overrightarrow a\) on \(\overrightarrow b\).

\(\overrightarrow a \cdot \overrightarrow b\) = \(|\overrightarrow a||\overrightarrow b|\) cos θ = \(|\overrightarrow b|\) OL

= \(|\overrightarrow b|\) (projection of \(\overrightarrow a\) on \(\overrightarrow b\))

Thus, projection of \(\overrightarrow a\) on \(\overrightarrow b = \dfrac{\overrightarrow a \cdot \overrightarrow b}{|\overrightarrow b|}\)

Similarly, the vector projection of \(\overrightarrow b\) on \(\overrightarrow a = \dfrac{\overrightarrow a \cdot \overrightarrow b}{|\overrightarrow a|}\)

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.

\(cos\theta = \dfrac{\overrightarrow a \cdot\overrightarrow b}{|\overrightarrow a|.|\overrightarrow b|}\)

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

\(\overrightarrow a \cdot \overrightarrow b\) = \((a_1 \hat i + a_2 \hat j + a_3 \hat k)(b_1 \hat i + b_2 \hat j + b_3 \hat k)\)

= \((a_1b_1) (\hat i. \hat i) + (a_1b_2) (\hat i.\hat j)+ (a_1b_3) (\hat i. \hat k) + \\(a_2b_1) (\hat j. \hat i) + (a_2b_2)(\hat j. \hat j) + (a_2b_3 (\hat j. \hat k) + \\(a_3b_1)(\hat k. \hat i) + (a_3b_2)(\hat k. \hat j) + (a_3b_3)(\hat k. \hat k)\)

= \(a_1b_1\) + \(a_2b_2\)+ \(a_3b_3\)

Note: This is because the dot product of two same unit vectors is 1 and of two different unit vectors is 0.

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. Then 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.

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.

• \(\overrightarrow a \cdot (\overrightarrow b+\overrightarrow c) = \overrightarrow a \cdot \overrightarrow b + \overrightarrow a \cdot \overrightarrow c\)
• \((\overrightarrow a+\overrightarrow b). \overrightarrow c = \overrightarrow a \cdot \overrightarrow c+ \overrightarrow b \cdot \overrightarrow c\)
• \(\overrightarrow a \cdot (\overrightarrow b - \overrightarrow c) = \overrightarrow a \cdot \overrightarrow b - \overrightarrow a \cdot \overrightarrow c\)
• \((\overrightarrow a -\overrightarrow b). \overrightarrow c = \overrightarrow a \cdot \overrightarrow c - \overrightarrow b \cdot \overrightarrow c\)

Nature of Dot Product

• We know that 0 ≤ θ ≤ π.
• If θ = 0 then \(\overrightarrow a \cdot \overrightarrow b\) = ab [Two vectors are parallel in the same direction ⇒ θ = 0 ] .
• If θ = π , \(\overrightarrow a \cdot \overrightarrow b\) = -ab [Two vectors are parallel in the opposite direction ⇒ θ = π.].
• If θ = π/2, then \(\overrightarrow a \cdot \overrightarrow b\) = 0 [Two vectors are perpendicular ⇒ θ = π/2]
• If 0 < θ < π/2, then cosθ is positive and hence \(\overrightarrow a \cdot \overrightarrow b\) is positive.
• If π/2 < θ < π then cosθ is negative and hence \(\overrightarrow a \cdot \overrightarrow b\) 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. |\(\overrightarrow a\)| = \(\sqrt{\overrightarrow a . \overrightarrow a}\)
• \(\overrightarrow a \cdot \overrightarrow a\) = \(|\overrightarrow a\)|² (or) it can be written as a²
• 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 \((\overrightarrow a \cdot\overrightarrow b)\)
• (\(\overrightarrow a - \overrightarrow b\)) ² = \(|\overrightarrow a\)|² + \(|\overrightarrow b|\)² - 2 \((\overrightarrow a \cdot\overrightarrow b)\)
• \((\overrightarrow a + \overrightarrow b). (\overrightarrow a - \overrightarrow b) = |\overrightarrow a\)|² - \(|\overrightarrow b|\)²

The dot product of the unit vectors is studied by taking the unit vectors \(\hat i\) along the x-axis, \(\hat j\) along the y-axis, and \(\hat k\) along the z-axis 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

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. \(\overrightarrow a \cdot \overrightarrow b\) = \(|\overrightarrow a||\overrightarrow b|\) cos 90º ⇒ \(\overrightarrow a \cdot \overrightarrow b\) = 0

Important Notes on Dot Product:

• The dot product or the scalar product of two vectors 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. \(\overrightarrow a \cdot \overrightarrow b\) = \(|\overrightarrow a|\overrightarrow b|\) 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 \(\overrightarrow a \cdot \overrightarrow b = a_1b_1 + a_2b_2+ a_3b_3\)

☛ Related Topics:

• Addition of Vectors
• Product of Vectors
• Types of Vectors

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. \(\overrightarrow a \cdot \overrightarrow b\) = \(a_1b_1\) + \(a_2b_2\)+ \(a_3b_3\).
• 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. ( \(\overrightarrow a \cdot \overrightarrow b\) = \(|\overrightarrow a||\overrightarrow b|\) \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 cos 0°= 1. Hence for two parallel vectors a and b we have \(\overrightarrow a \cdot \overrightarrow b\) = \(|\overrightarrow a||\overrightarrow b|\) cos 0° = \(|\overrightarrow a||\overrightarrow b|\).1 = \(|\overrightarrow a||\overrightarrow b|\).

What is the Difference Between Dot Product and Cross Product?

Here are the differences between the dot product and cross product of two vectors.

What is Dot Product Formula?

The dot product formula is if \(\overrightarrow a\) and \(\overrightarrow b\) are two vectors then their dot product is given by \(\overrightarrow a \cdot \overrightarrow b\) = \(|\overrightarrow a||\overrightarrow b|\) cos ⁡θ, where θ is the angle between the two vectors.

What are Dot Product Rules?

Here are two dot product rules which can be used depending on the available information:

• \(\overrightarrow a \cdot \overrightarrow b\) = \(a_1b_1\) + \(a_2b_2\)+ \(a_3b_3\).
• ( \(\overrightarrow a \cdot \overrightarrow b\) = \(|\overrightarrow a||\overrightarrow b|\) \cos θ).

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.

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×cos θ.

How To Find 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. (\(\overrightarrow a \cdot \overrightarrow b\) = \(|\overrightarrow a||\overrightarrow b|\) cos θ. Also check to dot product calculator, to easily find the vector dot product.

Why is the Dot Product Called Scalar Product of Two Vectors?

The dot product is a scalar because all the individual constituents of the answer are scalar values. In \(\overrightarrow a \cdot \overrightarrow b\) = \(|\overrightarrow a|.|\overrightarrow b|\).Cosθ, \(|\overrightarrow a|, |\overrightarrow b|\), and cos θ 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 cos 90° = 0, the dot product of two orthogonal vectors is equal to 0. \(\overrightarrow a \cdot \overrightarrow b\) = \(|\overrightarrow a| |\overrightarrow b|\) cos 90° = \(|\overrightarrow a| |\overrightarrow b|\) (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 \(\overrightarrow a \cdot \overrightarrow b\) = \(\overrightarrow b \cdot \overrightarrow a\), 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 non-zero vectors, the dot product is zero if the angle between the two vectors is 90º, because cos 90º = 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 their magnitudes (as cos 0 = 1). In fact, the cross product of two collinear vectors is a zero vector.