In geometric product we will basically deals with vectors product or vectors multiplication. So vectors are the quantity which are not fully described or indicated by a number of units alone. For example, both speed and direction are important to describe the course of an aircraft during flight. So we call a quantity , which has both magnitude and direction, a vector quantity. We can multiply vector with scalar quantity and we have more two types of multiplication related to geometric product one is dot product and another one cross product .
Types of Geometric Product
Basically there are two ways of multiply vectors .
Dot product :- This product is also known as scalar product and scaler product or dot product of two vactors can be represent as vector a dot vector b. Dot will be represent as ‘.’ And we can write it as a vector . b vector and it can be write as |a| x |b| x cos θ. Let me explain it by diagram. We will just multiply the magnitude of the vector a and b and will multiply cos θ and will get the the dot product of vector a and b. |a| represent the magnitude of vector a.
2. Cross product:- Cross product is also knwn as vector product and cross product of two vectors can be represent as vector a cross vector b. Cross will be represented as ‘x’ and we can cross product as vector a x vector b and this further will be simply as |a|× |b| × sin θ × n. let me explain it by using diagram. Here n is unit vector with magnitude as 1 and n will be perpendicular to both a and b. for cross product we just need to multiply the magnitude of a and b and then with sin θ.
So cross product of vector a and b is basically the area of parallelogram formed by vector a and b.
Problems on Geometric Product:
Q- what is the sign of dot product ?
Ans :- '.'
Q- How you will represent dot product of vector a and b ?
Ans :- vector a . b vector and final product will be |a| x |b| x cos `theta`.
Q- what is the sign for cross product ?
Ans :- 'x'
Q- what will be the resultant cross product of vector a and vector b ?
Ans :- |a| times |b| times sin `theta`
Q- Magnitude of vactor a and b are 2 and 5 and angle between them is 90 degree what will be cross product and dot product of both of them ?
Ans :- dot product will be zero and cross product will be 10.