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.
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.
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.
Scalar quantities vary when their magnitude changes.
A vector quantity is a quantity that has magnitude and is directed in a particular direction.
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.
Vector quantities change when the magnitude or direction or both change.
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
The dot product can be used for expressing the magnetic potential energy and the electric dipole potential.
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 =
Bx By Bz |
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
Bx By Bz |
= Ax Bx + Ay By + Az Bz
= A . B
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.
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,
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
i Ax Bx | j Ay By | k Az Bz | = |
A X B = i ( Ay Bz – Az By ) +j ( Az Bx – Ax Bz) + k ( Ax By – Ay Bx)
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.
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