Scalar Product: Meaning, Formula & Applications

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)

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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?

Answer:- 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.

Vector Quantity Definition

A vector quantity is a mathematical quantity that is defined by its magnitude and direction as two distinct qualities. The magnitude of the quantity with absolute value is represented here. In contrast, direction represents the north, east, south, west, north-east, and so on.

The vector quantity follows the triangle law of addition. A vector is represented by a vector quantity depicted by an arrow placed over or next to a symbol.

Difference between a Scalar and a Vector Quantity

A scalar quantity differs from a vector quantity when it comes to direction. Vectors have direction, whereas scalars do not. Because of this property, a scalar quantity is considered to be one-dimensional, whereas a vector quantity might be multi-dimensional. Let’s understand more about the differences between scalar quantity and vectors from the table below.

Important Differences Between Scalar Quantity and Vector Quantity

In terms of the scalar and vector difference, the following points are essential:

• The magnitude of a quantity is referred to as a scalar quantity. The vector quantity, on the other hand, considers both magnitude and direction to characterize its physical amount.

• Scalar quantities can describe one-dimensional numbers; for example, a speed of 35 km/h is a scalar quantity. Multidimensional values, such as temperature increases and decreases, can be stated using vector quantities.

• When only the magnitude changes, the scalar quantity changes; however, for the vector quantity, both the magnitude and direction must change.

• Scalar quantities conduct operations using standard algebra rules such as addition, multiplication, and subtraction, whereas vector quantities do operations using vector algebra rules.

A scalar quantity can also divide another scalar amount, however, two vector quantities cannot be divided.

Conclusion

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