Scalar and Vector Quantities

Introduction

In physics, there are many mathematical quantities that are used to define the motion of objects. These mathematical quantities are of two types and we are going to discuss their representation and the product of those quantities. Apart from that, we come across many terms like distance, displacement, acceleration, speed, and velocity.

Scalar and Vector Quantities

Scalar Quantities

A scalar quantity is a quantity that has no particular direction but has only magnitude or size. It has a numerical value and a unit.

Examples of Scalar Quantities

• Speed
• Distance
• Power
• Energy
• Time

Representation 

Let’s take the distance of 3 Km.

In this case, we only have the magnitude of distance as 3 but no direction is represented here. It also has the unit of distance Km.

Let us take another example of 9 ms-1

Here, the speed has a magnitude of 9 with no direction. Its unit is represented as ms-1.

Properties of a Scalar Quantity

Scalar quantities vary when their magnitude changes.

Vector Quantities

A vector quantity is a quantity that has magnitude and is directed in a particular direction.

Examples of Vector Quantities

• Velocity
• Displacement
• Acceleration
• Force
• Weight
• Momentum

Representation of Vector Quantities

If the acceleration 45 ms-2 at 60° with respect to the x-axis is considered, here the 45 ms-2 of acceleration occurs at the angle 60°. It has both magnitude and direction. It has its respective unit too.

Properties of a Vector Quantity

Vector quantities change when the magnitude or direction or both change.

Scalar Product

The product of the magnitude of two vectors is called a dot product or scalar product. The dot product is nothing but the product of the magnitude of two vectors and the cosine angles that they form with each other. It is also called a projection product or inner product.

For example,

A . B= |A| |B| cos

Where A  vector

|A|  magnitude of the vector

This is a scalar quantity that has a magnitude and no direction.

If the vectors are expressed in i ,j,  k  in x,y,z, direction respectively, like

A = Ax  i + Ay j+ Az k

B = Bx  i + By j + Bz k

The dot or scalar product is represented as

A . B = Ax Bx + Ay By + Az Bz

Applications of the Scalar Product

The dot product can be used for expressing the magnetic potential energy and the electric dipole potential.

Matrix Representation of the Scalar Product

If we represent the spatial vector B as column matrices, then A would be the transpose of it, as below.

A = [ Ax Ay Az ]

B = 

The product of these two vectors gives the single number as the sum of each product of spatial components of two vectors.

[ Ax Ay Az ]  and

=  Ax Bx + Ay By + Az Bz

= A . B

Vector Product

The product of two vectors is called a cross product or vector product. The cross product is nothing but the product of the two vectors and sine angle that they form with each other. It is also termed a directed area product.

The vector product is the product of magnitudes of two vectors with the sine of the angle between them, and a direction perpendicular to the plane.

C = A X B

If A and B are the two vector components, then the vector product would be A X B, defined by

A X B = |A| |B| sin z

 The vector product always produces another vector quantity.

z represents the direction that the vectors are perpendicular to each other. This is based on the right-hand screw rule, which is used to determine the direction of the vector product.

Right-hand Screw Rule

There would be a vector c perpendicular to the cross product of vectors A and B.

C  = A X B

This is explained by the right-hand screw rule. If you hold up the right hand,

•  the index finger is pointed towards the direction of A
• the middle finger is pointed towards the direction of B
• the thumb finger is pointed towards the direction of C

Properties of the Vector Product

• The vector product is the product of magnitudes of two vectors with the sine of the angle between them, and a direction perpendicular to the plane
• Vector products of two vectors are always a vector quantity
• The vector product of any two vectors would be always perpendicular to the two vectors
• The vector product of any two vectors is maximum when the vertices are orthogonal

Matrix Representation of the Cross Product

If we have the components of the vector A and B as

A = Ax  i + Ay j+ Az k

B = Bx  i + By j + Bz k

A X B = i ( Ay Bz – Az By ) +j ( Az Bx – Ax Bz) + k ( Ax By – Ay Bx)

Applications of Vector Product

• It is used to find the vector normal to a plane.
• It is used to find the angle between the two vectors
• It is used to calculate the moment of a force through a point
• It is used to find the moment of a force through a line

Difference of Scalar and Vector Products

• The scalar product gives a number as a result, but the vector products give a vector
• The scalar product works in any dimension, but the vector product works in 3 dimensions
• The vector product measures the two vector points in different directions, but the scalar product measures the two vector points in the same direction

Conclusion

In real life, scalars, and vectors are used for many applications. Scalars are used in math engines and scripting languages. Vectors are used in force and velocity calculation in space. In this article, we have clearly seen the equations of scalar and vector product and the representation of the matrix.