Multiplication of a vector by a real number or scalar is an important operation in vector algebra that changes the magnitude of the vector without altering its fundamental nature. When a vector is multiplied by a positive scalar, its magnitude increases or decreases proportionally while its direction remains unchanged. However, multiplication by a negative scalar changes both the magnitude and the direction of the vector. This concept is widely used in Physics and Mathematics for representing scaled vector quantities, vector components, and physical quantities such as force, displacement, velocity, and acceleration. It is an important topic in Class 11 Physics and forms a strong foundation for JEE Main, JEE Advanced, NEET, NDA, IMUCET, and other competitive examinations.
Multiplication of Vectors in Class 11 Physics
In mathematics and physics, vector multiplication involves multiplying a vector either by a scalar (a number) or another vector. The result depends on the type of multiplication: scalar multiplication stretches or shrinks the vector, while vector multiplication yields a new vector or a scalar based on the rule used.
Multiplication of vectors can be of two types :
(i) Scalar Multiplication
(ii) Vector Multiplication
Here, we will discuss only the Scalar Multiplication by a vector.
Understand related topics like What is the difference between scalar and vector quantities?
Multiplication of a Vector by a Real Number (n)
The multiplication of a vector $\vec{A}$ by a real number (n) becomes another vector n$\vec{A}$. Its magnitude becomes n times the magnitude of the given vector. Its direction is the same or opposite to that of given vector $\vec{A}$, according as n is a positive or negative real number. Thus,
n($\vec{A}$) = n $\vec{A}$ and –n($\vec{A}$) = –n$\vec{A}$.
For example, if a vector $\vec{A}$ is multiplied by a real number n = 2, we get 2$\vec{A}$, which is a vector, acting in the direction of vector $\vec{A}$ and having magnitude twice as that of vector $\vec{A}$.
If vector $\vec{A}$ is multiplied by real number n = – 2, then we get -2$\vec{A}$, which is also a vector, acting in the opposite direction of vector $\vec{A}$ and having magnitude twice as that of vector $\vec{A}$. The unit of vecor n$\vec{A}$ is the same as that of given vector $\vec{A}$.
If $\vec{A}$ = a $\vec{i}$ + b $\vec{j}$ + c $\vec{k}$, and n is a real number or scalar, then :
n$\vec{A} $ = (na)$\vec{i}$ + (nb)$\vec{j}$ + (nc)$\vec{k}$
This is also called scalar multiplication of a vector.
Example :
If $\vec{A}$ = 2$\vec{i}$ – 4$\vec{j}$ + 5$\vec{k}$, and n =-3, then :
-3$\vec{A} $ = -6$\vec{i}$ + 12$\vec{j}$ – 15$\vec{k}$
Multiplication of a Vector by a Scalar (S)
When a vector $\vec{A}$ is multiplied by a scalar S, it becomes a vector S$\vec{A}$, whose magnitude is S times the magnitude of vector $\vec{A}$ and it acts along the direction of vector $\vec{A}$. The unit of vector S$\vec{A}$, is different from the unit of vector $\vec{A}$.
For illustration, if $\vec{A}$ = 100 newton due west and S = 10 second, then S$\vec{A}$ = 10 second x 100 newton due west = 1000 newton-second due west.
From this discussion, the following points may be noted :
1. The negative of a vector is the result of multiplication of the vector with real number – 1. Thus,
– $\vec{A}$ = (-1)$\vec{A}$
2. The unit of the vector n$\vec{A}$ is the same as that of vector $\vec{A}$, if n is a pure real number or a dimensionless scalar. In other words, the unit of a vector does not change, when it is multiplied by a dimensionless scalar.
3. The unit of the vector n$\vec{A}$ is totally different from that of vector $\vec{A}$, if n is a dimensional scalar.
For example, if n is time and the vector $\vec{A}$ is velocity, the units of vector n$\vec{A}$ will be that of the displacement; and if n is mass and vector $\vec{A}$ is acceleration, the unit of vector n$\vec{A}$ will be that of the force.
Note : Vector multiplication yields a new vector based on the rule used.
Build strong concepts by studying Scalars and Vectors Class 11 Physics Notes With Detail Explanation
Addition of Vectors
Since vectors have both magnitude and direction they cannot be added by the method of ordinary algebra. Thus, vectors can be added geometrically or analytically using certain rules called ‘vector algebra’. In order to find the sum (resultant) of two vectors, which are inclined to each other, we use
(i) Triangular law of addition method or
(ii) Parallelogram law of vectors.
This we will discuss in next articles.
| Noteworthy Point |
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| In standard vector algebra, dividing one vector by another vector is generally not defined because there is no unique way to perform the operation. |
Conceptual Short Questions and Answers Based on Multiplication of Vector by Real Number and Scalar
What is meant by multiplication of vectors?
Multiplication of vectors refers to the process of multiplying a vector either by a scalar or by another vector. The result obtained depends on the type of multiplication being performed.
What are the different types of vector multiplication?
Vector multiplication is broadly classified into two types: scalar multiplication and vector multiplication. In scalar multiplication, a vector is multiplied by a scalar, while vector multiplication involves operations between two vectors.
What happens when a vector is multiplied by a positive real number?
When a vector is multiplied by a positive real number, its magnitude increases or decreases proportionally according to the value of the number, while its direction remains unchanged.
What happens when a vector is multiplied by a negative real number?
Multiplication by a negative real number changes the magnitude of the vector according to the numerical value and reverses its direction.
Does scalar multiplication change the nature of a vector?
No, scalar multiplication does not change the vector nature of a quantity. The result remains a vector, although its magnitude and sometimes its direction may change.
Does the unit of a vector always remain the same after multiplication?
The unit remains unchanged when a vector is multiplied by a dimensionless scalar. However, when multiplied by a dimensional scalar, the resulting vector may have different units.
Why is scalar multiplication important in Physics?
Scalar multiplication is widely used to represent scaled physical quantities, modify vector magnitudes, and solve problems involving displacement, velocity, acceleration, force, and momentum.
Can a vector be multiplied by zero?
Yes, multiplying a vector by zero results in a zero vector, which has zero magnitude and no specific direction.
How is the negative of a vector obtained?
The negative of a vector is obtained by multiplying the vector by –1. The resulting vector has the same magnitude but acts in the opposite direction.
Can one vector be divided by another vector?
In standard vector algebra, division of one vector by another vector is generally not defined because there is no unique mathematical operation for vector division.