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The simple algebraic procedure cannot be used to multiply two vector quantities. A scalar or a vector can be formed by multiplying two vectors. Both ‘force’ and ‘displacement,’ are vector quantities. Their output could be ‘work’ or moment of force.’ Work is a scalar quantity, but the moment of force is a vector.
Vectors are used to represent vector quantities. If the product of two vectors is a scalar quantity, it is referred to as a scalar product, ‘but if the product is a vector quantity, it is referred to as a vector product.’ If A and B are two vectors, the scalar product is A.B, and the vector product is A xB. As a result, the scalar product is also known as the ‘dot product,’ while the vector product is also known as the ‘cross-product.’
Vectors and Scalars
Quantities in physics are classified as scalars or vectors. The main distinction is that a vector is related with a direction, whereas a scalar is not. A scalar quantity has only one magnitude. It is specified by a number, together with the appropriate unit. The distance between two points, the mass of a thing, the temperature of a body, and the time at which an event occurred are all examples. Ordinary algebra provides the principles for combining scalars. Scalars can be multiplied, divided, and added to.
A vector quantity has both a magnitude and a direction and obeys the triangle or parallelogram law of addition. As a result, a vector is defined by specifying its magnitude as a number and its direction as a direction. Vectors can represent a variety of physical properties, including displacement, velocity, acceleration, and force. A vector is commonly represented by an arrow placed above a letter, such as v because boldface is difficult to construct when written by hand. As a result, the velocity vector is represented by both v and v. The absolute value of a vector, denoted by |v| = v, is often referred to as its magnitude.
Multiplication of Vector quantity
Vector multiplication is one of the simple and important ideas in mathematics. We use the multiplication of vectors to find a product of any two vectors whether it is a scalar or as a vector. There are two forms of vector multiplications: one is dot product and another is cross product. The magnitude of a vector is multiplied by a number then it is called a vector is multiplied by a scalar.
Using Scalars to Multiply Vectors
Even though vectors and scalars represent different types of physical quantities, they must sometimes interact. Because of their different dimensions, adding a scalar to a vector quantity is nearly impossible. A vector quantity, on the other hand, can be multiplied by a scalar. The inverse, on the other hand, is not possible. A scalar can never be multiplied by a vector, for example.
Similar quantities are subjected to arithmetic multiplication when vectors and scalars are multiplied. In other words, the magnitude of vectors is multiplied by the magnitude of scalar quantities. A vector is a result of multiplying vectors with scalars. The product vector has the same direction as the vector multiplied by the scalar, and its magnitude is increased by the same number of times as the product of the vector and scalar magnitudes.
Multiplication of Vectors using Scalars in Real-World Applications
In physics, vector multiplication with scalars has a wide range of applications. Many SI units of vector values are the vector and scalar products. The SI unit of velocity, for example, is the meter per second. The term “velocity” refers to a vector quantity. This is calculated by multiplying two scalar values, length and time, with a unit vector in one direction. There are numerous other applications of vector multiplication with a scalar in mathematics and physics.
Multiplication of Vectors with real number
When you multiply a vector A by a positive number, you get a vector whose magnitude is modified by the factor but whose direction is unchanged: A = A if > 0. When A is multiplied by 2, for example, the resultant vector 2A is in the same direction as A and has twice the magnitude of |A|.
When you multiply a vector A by a negative number, you get another vector with the opposite direction as A and the same length.
|A| is the magnitude of vector A. When a given vector A is multiplied by negative numbers, The factor multiplied by a vector A could be a scalar with its physical dimension. The dimension of A is then equal to the sum of the dimensions of and A. For example, we can generate a displacement vector by multiplying a constant velocity vector by duration (of time).
Table showing multiplication of vector with a real number with 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 as 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.
Example of Scalar Vector Multiplication
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 described 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 in physics of vector multiplication with scalars.
Conclusion
In this post, we discussed the multiplications of vectors with real numbers and with other vectors. We found that the multiplication of a vector with a real number is called scalar multiplication whereas vector multiplication is a composition of linear transformations. The only way to multiply a vector by a real number is to combine the scaling factor with the original vector, placing it all inside of the magnitudes. In conclusion, we see that the multiplication of vectors with real numbers is one of the most useful mathematical tools.
