Multiplication of Vectors by a Real Number

Scalar and vector quantities are just the two kinds of quantities.

If we want to measure any quantity, there are two ways of measuring either by only magnitude or by both magnitude and direction.

If a quantity is measured by its magnitude, it is called a scalar quantity.

If a quantity is measured by both magnitude and direction, it is called a vector quantity.

This article will look at the multiplication of vectors by a real number.

A vector is a quantity that possesses both magnitude and direction but does not have a position.

Velocity and acceleration are two examples of such numbers. Vector analysis was invented independently by Josiah Willard Gibbs and Oliver Heaviside (of the United States and the United Kingdom, respectively) late in the nineteenth century to explain the new principles of electromagnetic theory discovered by Scottish scientist James Clerk Maxwell. Since then, vectors have been indispensable in physics, mechanics, electrical engineering, and other sciences for quantitatively describing forces.

It is possible to add or remove two vectors. For example, to add or remove vectors v and w  graphically, move them both to the origin and complete the parallelogram produced by the two vectors; v  + w  is one diagonal vector of the parallelogram.

Multiplying two vectors can be done in two different ways. Dot product (a・b ) and cross product(a×b)

The output of the cross product is another vector. The right-hand rule can be used to depict the direction of v×w, which is perpendicular to both v and w . The cross product is commonly used to obtain a “normal” (a line perpendicular) to a surface at some point, and it is also used to calculate torque and magnetic force on a moving charged particle.

A dot product is another technique for multiplying two vectors together. v・w  = vw cos θ gives the dot product. 

The work W performed by a constant force F acting on a moving object d is given by W = Fdcosθ in a typical physical application.

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Multiplication of Vectors

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 term “vector multiplication” refers to one of several ways for multiplying two (or more) vectors with each other. It could be about one of the following articles:

• Dot product — a binary operation that accepts two vectors and gives output as scalar amount, sometimes known as the “scalar product.” The dot product of two vectors is the product of the magnitudes of the two vectors and the cosine of the angle between them.

As a result, A・B   = |A | |B | cos θ

In general, a bilinear product in an algebra over a field.

• The cross product, also called as the “vector product,” is a binary operation on two vectors that yields another vector. In 3-space, the vector perpendicular to the plane determined by the cross product of two vectors is defined as the vector whose magnitude is the product of the magnitudes of the two vectors plus the sine of the angle between the two vectors.

 As a result, A×B  = |A | |B | sin θ n and where n  is the unit vector perpendicular to the plane determined by vectors A  and B.

• A matrix is the outer product of two coordinate vectors. The outer product of two vectors with n and m is an m matrix. In general, the outer product of two vectors (multidimensional arrays of numbers) is a vector. The cross product, also known as the outer product of vectors.

• Products with more than three vectors are known as multiple cross products.

There are two types of vector multiplication:

• Multiplication of Scalars

•  Multiplication of Vectors

Few examples of multiplication of vectors with scalar :

1. Consider the case when a vector, let’s call it ‘a,’ is multiplied by a scalar with a magnitude of 0.25. In this situation, the product vector indicates a vector with the same direction as vector a .

2. Force is a vector quantity in the physical world. The magnitude determines the amount of work done. 

And the 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. F = ma is the formula for the force. In the equation above, ‘a’ signifies the object’s acceleration and ‘m’ denotes the object’s mass, a scalar quantity. As a result, it is one of the physics examples of vector multiplication with scalars.

Vector Multiplication

Multiplication of two vectors is more complicated than scalar multiplication. The scalar product is sometimes known as the “dot product” and the vector product is the way of multiplication that uses two vectors (or “cross product”).

We’ll only talk about the scalar product for now, but you should have a good enough mathematical background to comprehend the vector product as well. In two dimensions, the scalar product (or dot product) of two vectors is defined. This definition, as always, can readily be extended to three dimensions by just following the pattern. To distinguish the operation from the vector product, which uses a cross sign (), the operation should always be denoted with a dot (•)—hence the terms dot product and cross product.

Conclusion:

When a vector is multiplied by a scalar, the outcome is a larger-magnitude vector with the same direction as the original.

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.