PROVE THAT THE SCALAR PRODUCT OBEYS THE DISTRIBUTIVE LAW

If we have three vectors A, B and C (as shown in the figure) then according to the distributive law we have

A.(B + C) = A.B + A.C

Now from the physical meaning of the Scalar Product

A.B = |A|.|projection of B on vector A| 

A.C = |A|.|projection of C on vector A| 

so, from the above figure

A.B = |A|OP

A.C = |A|PQ

or we have

A.B + A.C = |A|OP + |A|PQ

or

A.B + A.C = |A|(OP + PQ)

from teh figure

A.B + A.C = |A|OQ

similarly,

A.(B + C) = |A|.|projection of (B+C) on vector A|

or from the above figure 

A.(B + C) = |A|OQ

thus, we conclude

A.(B + C) = A.B + A.C