Scalar Product

Define Scalar Product

Most of the quantities that we know are generally classified as either a scalar quantity or a vector quantity. There is a distinct difference between scalar and vector quantities. Scalar quantities are among those quantities where there is only magnitude, and no direction. Their results can be calculated directly. 

For vector quantities, magnitude and direction, both must be available. Hence, the result calculated will also be based on the direction. One can consider displacement, torque, momentum, acceleration, velocity, and force as a vector quantity. 

When it comes to calculating the resultant of vector quantities, then two types of vector product can arise. One is true scalar multiplication, which will produce a scalar product, and the other will be the vector multiplication where the product will be a vector only. 

In this article, we will discuss the scalar product in detail.


Scalar Product of Two Vectors

The Scalar product is also known as the Dot product, and it is calculated in the same manner as an algebraic operation. In a scalar product, as the name suggests, a scalar quantity is produced.

Whenever we try to find the scalar product of two vectors, it is calculated by taking a vector in the direction of the other and multiplying it with the magnitude of the first one. If direction and magnitude are missing, then the scalar product cannot be calculated for vector quantity.

To understand it in a better and detailed manner, let us take an example-

Consider an example of two vectors A and B. The dot product of both these quantities will be:-

\[\widehat{A}\] . \[\widehat{B}\] = ABcos𝜭

Here, θ is the angle between both the vectors.

For the above expression, the representation of a scalar product will be:-       

\[\widehat{A}\] . \[\widehat{B}\] = ABcos𝜭 = A(Bcos𝜭) = B(Acos𝜭)

(Image to be added soon)      

We all know that here, for B onto A, the projection is Bcosα, and for A onto B, the projection is Acosα. 

Now, we can clearly define the scalar product as the product of both the components A and B, along with their magnitude and their direction. For the product of vector quantities, it is important to get the magnitude and direction both.


Commutative Law

Commutative law is related to the addition or subtraction of two numbers. This law is also applicable to scalar products of vectors. This property or law simply states that a finite addition or multiplication of two real numbers stays unaltered even after reordering of such numbers. This goes with the vectors also. The result of a scalar product remains unchanged even after the reordering of vectors while extracting their product. 

\[\widehat{A}\] . \[\widehat{B}\] = \[\widehat{B}\] . \[\widehat{A}\]


Distributive Law

The distributive law simply states that if a number is multiplied by a sum of numbers, the answer would be the same if such number would have been multiplied by these numbers individually and then added. This distributive law can also be applied to the scalar product of vectors. For better understanding, have a look at the example below-

\[\widehat{A}\] . ( \[\widehat{B}\] + \[\widehat{C}\] ) = \[\widehat{A}\] . \[\widehat{B}\] + \[\widehat{A}\] . \[\widehat{C}\]

\[\widehat{A}\] . λ \[\widehat{B}\] = λ (\[\widehat{B}\] . \[\widehat{A}\])

Here, λ is the real number.

After understanding the commutative law and distributive law, we are ready to discuss the dot product of two vectors available in three-dimensional motion.

All of the three vectors should be represented in the form of unit vectors.

\[\widehat{A}\] - Axi

\[\widehat{A}\] = Axi + Ayj + Azk

\[\widehat{B}\] = Bxi + Byj + Bzk

Here,

For X- Direction the unit vector is i

For Y- Direction the unit vector is j

For Z- Direction the unit vector is k

Now, when it comes to looking at the scalar product of all these two factors, it will be given by:-

 \[\widehat{A}\] . \[\widehat{B}\] = (Axi + Ayj + Azk) . (Bxi + Byj + Bzk)

\[\widehat{A}\] . \[\widehat{B}\] = AxBx + AyBy + AzBz

Here,

\[\widehat{i}\] . \[\widehat{i}\] = \[\widehat{j}\] . \[\widehat{j}\] = \[\widehat{k}\] . \[\widehat{k}\] = 1

\[\widehat{i}\] . \[\widehat{j}\] = \[\widehat{j}\] . \[\widehat{k}\] = \[\widehat{k}\] . \[\widehat{i}\] = 0


Solved Examples

Question :- There is a force of F = (2i + 3j + 4k) and displacement is d = (4i + 2j + 3k), calculate the angle between both of them?

Ans:- We know, A.B = AxBx + AyBy + AzBz

Thus, F.d= Fxdx + Fydy + Fzdz 

= 2*4 + 3*2 + 4*3 

= 26 units

Alternatively,

F.d= F dcosθ

Now, F² = 2² + 3² + 4²

= √29 units

Similarly, d² = 4² + 2² + 3²

= √29 units

Thus, F d cosθ = 26 units.

FAQ (Frequently Asked Questions)

1. Is It Important For Vector Quantities To Have Both Magnitude And Direction?

Yes, vectors are called vectors because they have both magnitude and direction. If in case, only magnitude is there, and no direction, then the quantity will be considered as the scalar quantity. To calculate the difference between both the quantities, one has a look at the representation. For example, if there is a vector with magnitude 4 and direction along the x-axis, it will be represented as 4i, and if it is a scalar quantity, then it will be represented as 4. The representation of quantities will help you to understand whether you are dealing with a scalar quantity or a vector quantity.

2. Will I Be Able To Calculate The Product Of Vector Quantities Directly?

No, you cannot calculate the product of the vector quantities directly. The product of two vectors can be a complicated one as it can produce either a scalar or a vector quantity. There are many things that come into play while extracting the product, such as the direction of the cross product, which can be found using the right-hand thumb rule. Also, multiple laws are available like commutative law, distributive law, and others that will help an individual to calculate the product easily.