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Scalar and vector quantities are just the two kinds of quantities. If we want to measure any quantity, there are two ways to measure either by only magnitude or magnitude and direction. If a quantity has only magnitude, it is called a scalar quantity. If a quantity has both magnitude and direction, it is called a vector quantity. This article will look at the multiplication or division of a vector by a scalar.
Multiplication of a Vector by a Scalar
If is a scalar and a vector, their product will be denoted as a.
According to the value of the (either positive or negative), the new vector a will have the same direction as vector a or opposite.
The magnitude of the new vector a is the magnitude of scalar times the magnitude of the vector. a = a
If = 1a, then a = a = 1aa = 1
However, provided that vector a 0. Therefore, a shouldn’t be a null vector for the above condition to hold.
Distributive laws for addition and multiplication of a vector by a scalar-
Let k and m be any scalars and a and b be any two vectors.
ka + ma = (k + m) a
k(ma) = (km)a
k (a + b) = ka + kb
For example, vector a = i – 2j with magnitude 7 units.
The unit vector in the direction of vector a = 1a a = 15 (i – 2j) = 15i – 25j
The new vector of magnitude 7, which is in the direction of vector a is
7 a = 7 (15i – 25j) = 75i – 145j
How to Multiply the Vector by a Scalar?
To multiply the vector by a scalar is very easy; multiply the magnitudes of the vector with the scalar.
When we multiply a vector A by a real number k, we receive another vector A’. The magnitude of A’ is k times the magnitude of A. If it is positive, the direction of A’ is the same as the direction of A .
The resultant new vector will have the same direction.
For example, let c = ⟨8⟩. Find 7c.
7c = 7⟨8⟩
= ⟨7(8)⟩
= ⟨56⟩
Scalar Multiple of a Vector
When a vector is multiplied by a scalar, the magnitude of the vector changes.
The direction of the scalar multiple remains the same as the input vector.
Either the scalar is zero or the vector is zero if the scalar multiple of a vector is zero.
Table showing the multiplication of vector with a real number with a different factor:
Factors | Original vector | Product of vector after multiplication | The direction of the vector after multiplications |
If λ is greater than 0 | A | λA | Same of A |
– λ less than 0 | A | λA | Opposite to A |
λ equal to 0 | A | 0 = null vector | None. The starting and ending positions are the same. |
In general, the multiplication of a real number r by a vector A produces this result. It’ll provide us something in the same direction as before, but with r times the original magnitude. r A has two components: r Ax and r Ay in terms of components.
If r is equal to -1, the resultant vector will point in the opposite direction if A is true.
An Example of Multiplication or Division of a Vector by a Scalar
Force is a vector quantity in the physical world. The amount of work done is determined by the magnitude and direction of the force applied to the item. According to Newton’s second law of linear motion, this force is a product of a vector and a scalar number.
The force is as follows:
m x a = F.
In the equation above, a is the object’s acceleration, which is a vector quantity and m denotes the object’s mass, which is a scalar quantity. It is one example of the physics of vector multiplication with scalars.
Conclusion
The vector is a quantity that is referred to as having both direction and magnitude. Scalar is the quantity that consists of only the magnitude when both the vector and scalar quantity are multiplied. We know the magnitude of both vector and scalar as a product and the single direction that we attain through the vector quantity. The scalar quantity is denoted as while the vector is denoted with a and the product of both will be a. The product can either be positive or negative. The vector and scalar quantities are easy to calculate and decipher.
