define the scalar and vector product of two vectors a and b. if for three non zero vectors a, b and c, a.b=a.c and a*b=a*c then show that b=c.

Scalar product is a special technique of multiplying two vectors. As the name suggests, we get a scalar quantity after multiplication. Scalar product is also known as dot product (since it is denoted by a dot between the vectors) or inner product. This is an operation which intakes two vector quantities and returns a single scalar quantity after the multiplication. The scalar product can be found either geometrically or algebraically

a.b= ab cos<theta>
a.c=ac cos<phi>
cos<theta>= c/b cos<phi>............. (1)
similarly
Ia*bI= ab sin<theta>
Ia*cI= ac sin<phi>
sin<theta>=c/b sin<phi>.................. (2)
on squaring and adding (1) and (2), we get
1= c^2/b^2
c^2= b^2
so c=b.......... [ans}