Properties Of Scalar Product Of Two Vectors

A scalar product can be defined as the product of the magnitudes of two vectors and the cosine of the angle between them or the sum of the products of the corresponding entries of two sequences of numbers. Based upon the types of products for vectors, there are various applications of them in geometry, mechanics, and engineering. This article will give you in-depth knowledge about the scalar product of two vectors, its properties and their applications.

Key Terms: Scalar Product, Magnitude, Geometry, Vectors, Scalar Triple Product of Vectors, Angle, Magnitude, Cosine

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Definition of Scalar Product

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The scalar product of two vectors can be defined as the product of magnitudes of two vectors and the cosine of the angle between them.

Scalar Product

Let \(\overrightarrow{a}\) and b be two non zero vectors having a magnitude of |a| and |b| respectively and the angle between them is θ as shown in the figure above. According to the definition the scalar product of a and b is represented as 

\(\overrightarrow{a}\) . b = |a| |b|cosθ, where 0 ≤ θ ≤

If either \(\overrightarrow{a}\) or b are equal to zero then, θ is not defined. In such a case, the product of two vectors, a and b is equal to zero. 

If \(\overrightarrow{a}\) = 0 or b = 0, then \(\overrightarrow{a}\) . b = 0

Let us consider an example to know more about the scalar product of vectors.

Let us assume two vectors A and B. Let A has a modulus of 8 units, B has a modulus of 6 units, and the angle between them is 60 degrees.

Then the scalar product of vector A and vector B is 

\(\overrightarrow{A}\). B = |A | |B|cosθ

= 8 x 6 x cos60°

= 8 x 6 x 12

= 24

From the above example, you can see that product of two vectors is a real number, which is a scalar, and not a vector. So this is how we can combine two vectors to produce a scalar quantity.

Scalar Triple Product of Vectors

If \(\overrightarrow{a}\), \(\overrightarrow{b}\), and \(\overrightarrow{c}\) are three vectors. Then, the scalar product of \(\overrightarrow{a}\) and \(\overrightarrow{b}\) x \(\overrightarrow{c}\) is called a scalar triple product of \(\overrightarrow{a}\).\(\overrightarrow{b}\) and \(\overrightarrow{c}\) and it is denoted by [\(\overrightarrow{a}\) \(\overrightarrow{b}\) \(\overrightarrow{c}\)]

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Some Important Points About Scalar Product

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• The angle between two nonzero vectors is given by

cosθ = \(\frac{\overrightarrow{a}.\overrightarrow{b}}{|\overrightarrow{a}||\overrightarrow{b}|}\), or θ = cos^{-1(\(\frac{\overrightarrow{a}.\overrightarrow{b}}{|\overrightarrow{a}||\overrightarrow{b}|}\))}

Read More: Angle Between Two Vectors

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Properties of Scalar Product

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Property 1: Distributive of scalar product over addition

The scalar product follows the distributive law that implies, for any three vectors a, b, and c, then

• \(\overrightarrow{a}\). ( \(\overrightarrow{b}\) + \(\overrightarrow{c}\) ) = \(\overrightarrow{a}\) . \(\overrightarrow{b}\) + \(\overrightarrow{a}\) . \(\overrightarrow{c}\)

Property 2: Scalar product of two vectors is commutative

For any two vectors \(\overrightarrow{a}\), and \(\overrightarrow{b}\), the scalar product of them is commutative i.e, the order of the vectors in the product does not matter. 

• \(\overrightarrow{a}\) . \(\overrightarrow{b}\) = \(\overrightarrow{b}\) . \(\overrightarrow{c}\)

Why scalar product of two vectors is commutative?

Let A and B be two vectors and θ be the angle between them. 

According to the definition of the scalar product 

A.B = AB cosθ > 1

and B.A = BAcosθ > 2

Comparing Eq 1 and Eq 2 

We get A.B = B.A 

Property 3: If \(\overrightarrow{a}\), and \(\overrightarrow{b}\) are two vectors, then the scalar product of \(\overrightarrow{a}\), and \(\overrightarrow{b}\) and the vectors themselves are always a real number.

Property 4: If \(\overrightarrow{a}\), and \(\overrightarrow{b}\) are two vectors, then the scalar product of \(\overrightarrow{a}\) and \(\overrightarrow{b}\) is equal to zero if and only if \(\overrightarrow{a}\), and \(\overrightarrow{b}\) are perpendicular to each other. 

\(\overrightarrow{a}\) . \(\overrightarrow{b}\) = 0, if \(\overrightarrow{a}\) ? \(\overrightarrow{b}\)

Property 5: If \(\overrightarrow{a}\), and \(\overrightarrow{b}\) are two vectors, then the angle between two nonzero vectors is given by

cosθ = \(\frac{\overrightarrow{a}.\overrightarrow{b}}{|\overrightarrow{a}||\overrightarrow{b}|}\), or θ = cos^{-1(\(\frac{\overrightarrow{a}.\overrightarrow{b}}{|\overrightarrow{a}||\overrightarrow{b}|}\))}

Property 6: If \(\overrightarrow{a}\), and \(\overrightarrow{b}\) are any two vectors, and let 1 be any scalar. Then 

(λ\(\overrightarrow{a}\)). b = (λ\(\overrightarrow{a}\)). b = λ(\(\overrightarrow{a}\). \(\overrightarrow{b}\)) = a. (λb)

Thus \(\overrightarrow{a}\) . \(\overrightarrow{b}\) = a₁b₁ + a₂b₂ + a₃b₃

Property 8: For any two non zero vectors, a, and b, if θ = 0, then \(\overrightarrow{a}\) . b = |\(\overrightarrow{a}\)| |b|

In such a case, \(\overrightarrow{a}\) . \(\overrightarrow{a}\) = |\(\overrightarrow{a}\)|2, as θ = 0

Property 9: For any two non zero vectors, a, and b, if θ = π, then a . b = -|\(\overrightarrow{a}\)| |b|

In such a case,\(\overrightarrow{a}\). \(\overrightarrow{a}\) = |\(\overrightarrow{a}\)| |\(\overrightarrow{a}\)|, as θ = π

Property 10: Considering properties 4 and 8, it can be stated that for mutually perpendicular unit vectors i, j, and k, we have

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Things To Remember

• The scalar product is also called dot product and involves only the magnitude of the vectors. The dot product of Euclidean vectors a and b is 

 \(\overrightarrow{a}\) . b = |a| |b|cosθ, where 0 ≤ θ ≤

• The algebraic definition of the scalar product of two vectors can be written as a = [a₁, a₂, …, a_n] and b = [b₁, b₂, …, b_n] is defined as

a . b =i=\(\displaystyle\sum_{i=0}^{n}\)=a₁b₁+a₂b₂++a_nb_n 

• If vectors are identified with row matrices, the dot product can also be written as a matrix product 

a . b = ab^T

where b^T denotes the transpose of b

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