The document proves that u . cv = c(u . v) for all vectors u and v in R^n and all scalars c. It considers two cases: when c is positive and when c is negative. In both cases, it shows that u . cv equals c(u . v) by manipulating the equations. Therefore, the relationship u . cv = c(u . v) holds true for any vectors u and v and scalar c.

1. Prove that u . cv = c(u . v) for all vectors u and v in R" and all scalars c.
Solution
u.cv = |u|*|cv|*cos?
where ? = angle between u and cv
let a be the angle between u and cv when c is positive then
when c is negative the angle between u and cv will be p-a
case 1 : c is +ve ( |c|= c )
u.cv = |c|*|u|*|v|*cosa
=c*|u|*|v|*cosa
=c(u.v)
case 2 : c is -ve ( |c| = -c)
u.cv = |c|*|u|*|v|*cos(p-a)
=(-c)*|u|*|v|*(-cosa)
=c*|u|*|v|*cosa
= c(u.v)
hence u.cv = c(u.v)
when c is negative the angle between u and cv will be p-a
case 1 : c is +ve ( |c|= c )
u.cv = |c|*|u|*|v|*cosa
=c*|u|*|v|*cosa
=c(u.v)
case 2 : c is -ve ( |c| = -c)

2. u.cv = |c|*|u|*|v|*cos(p-a)
=(-c)*|u|*|v|*(-cosa)
=c*|u|*|v|*cosa
= c(u.v)
hence u.cv = c(u.v)
u.cv = |c|*|u|*|v|*cos(p-a)
=(-c)*|u|*|v|*(-cosa)
=c*|u|*|v|*cosa
= c(u.v)
hence u.cv = c(u.v)
u.cv = |c|*|u|*|v|*cos(p-a)
=(-c)*|u|*|v|*(-cosa)
=c*|u|*|v|*cosa
= c(u.v)
hence u.cv = c(u.v)