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A number is always the measured value when we measure a physical quantity. This number makes sense only when the corresponding unit is supplied. As a result, a measurement’s result has a numerical value as well as a unit of measurement.
A body’s mass, for example, is 3 kilogrammes. A quantity with the numerical value 3 and the unit of measurement kg are used in this example. The Magnitude’ is the numerical value combined with the unit. For the comprehensive description of some physical quantities, only magnitude is required. Magnitudes include body mass, distance between two places, time, temperature, height, the number of pendulum swings, and the number of books in a bag. They don’t have any sense of direction.
Vector multiplication with a scalar quantity
When a vector is multiplied by a scalar quantity, the magnitude of the vector changes in proportion to the magnitude of the scalar, but the direction of the vector remains unchanged.
Vectors and Scalars are multiplied
Despite the fact that vectors and scalars represent different types of physical qualities, it is often important for them to interact with one another. Because of the disparities in dimensions between a scalar and a vector quantity, it is virtually difficult to combine them. A vector quantity, on the other hand, can be multiplied by a scalar quantity. At the same time, it is not possible to achieve the opposite result. As a result, a scalar can never be multiplied by a vector in any way.
When vectors and scalars are multiplied together, the quantities that are similar to one another are subjected to arithmetic multiplication. In other words, the magnitude of vectors is multiplied by the magnitude of scalar quantities. Adding vectors to scalars yields a vector as the result of the multiplication operation. The direction of the product vector is the same as the direction of the vector that is multiplied with the scalar, and its magnitude is increased by the number of times the product of the magnitudes of the vector and scalar that are multiplied by each other.
Consider the following scenario: we have a vector a. This vector is multiplied by a scalar quantity k, and the result is another vector with a magnitude equal to ka and the vector’s direction remains the same as before a. k can be either positive or negative, and when k is either positive or negative, the direction of k becomes simply opposite of the vector’s direction.
For the time being, let us visually illustrate the scalar multiplication of the vector.
Assume that the values of ‘k’ are = 2,3,-3,-½ and so on.
and so forth.
We can observe from the collection of vectors presented above that the direction of each vector is a.
When the value of the scalar is positive, the direction of the arrow remains unchanged; when the value of the scalar is negative, the direction of the arrow becomes exactly opposite; and in both circumstances, the magnitude of the arrow changes based on the values of the scalar multiple.
In light of the foregoing discussions, we may observe that ka= |k|a.
Let us suppose that the value of the scalar multiple k is -1, and we know that the resultant vector is from scalar multiplication that it is a+(-a) = 0.
The vector -a symbolizes the negative or additive inverse of a vector.
Consider the following scenario: the value of k =1/|a| considering the fact that the value of a not equal to zero.
In this case, we have the property of a scalar multiple vectors.
ka is equal to |k|a=1/| a|xa
Also, as previously discussed, if k = 0, the vector is reduced to zero as well.
Consider the following illustration to better understand what I’m trying to say:
As an illustration, a vector is represented as follows in the orthogonal system:
a = 3i + j + k
What would be the resultant vector if this were to happen?
If the answer is multiplied by 5?
Due to the fact that the vector is to be multiplied by the scalar, the resulting product is
5a = 5(3i + j + k)
Or, 5a = 15i + 5j + 5k
Properties of Scalar Multiplication
Taking the absolute value of the scalar and multiplying it by the magnitude of the vector, the magnitude of the scaled vector is obtained. | ||CV||=|C|V |
Associative Property | c(du)=(cd)u |
Commutative Property | cu=uc |
Distributive Property | (c+d)u=cu+du c(u+v)=cu+cv |
Identity Property | 1.u=u |
Multiplicative Property of −1 | (-1)c=-c |
Multiplicative Property of 0 | 0(u)=0 |
Multiplication of vectors with scalars has a variety of practical applications
It is possible to find a wide variety of applications for the multiplication of vectors with scalars in the field of physics. Many of the SI units of vector values are the products of the vector and the scalars, as seen in the table below. Taking velocity as an example, the SI unit of velocity is the metre per second. The quantity of velocity is a vector quantity. In order to reach this result, multiply the two scalar values of length and time by a unit vector pointing in a given direction. There are numerous other cases in mathematics and physics in which vector multiplication with a scalar is employed to solve problems.
Conclusion
Despite the fact that vectors and scalars represent different types of physical qualities, it is often important for them to interact with one another. Because of the disparities in dimensions between a scalar and a vector quantity, it is virtually difficult to combine them. A vector quantity, on the other hand, can be multiplied by a scalar quantity. When vectors and scalars are multiplied together, the quantities that are similar to one another are subjected to arithmetic multiplication. It is possible to find a wide variety of applications for the multiplication of vectors with scalars in the field of physics. Many of the SI units of vector values are the products of the vector and the scalars, as seen in the table below. Taking velocity as an example, the SI unit of velocity is the metre per second.
