Whatever the angle, this dot product of v / II v II with w / II w II never exceeds one. That
is the "Schwarz inequality" Iv • wi < Ilvllllwll for dot products-or more correctly the
Cauchy-Schwarz-Buniakowsky inequality.
It was found in France and Germany
and Russia (and maybe elsewhere-it is the most important inequality in mathematics).
Since I cos () I never exceeds 1, the cosine formula gives two great inequalities:
'S0HWARZINEQUi\:LI:'l'Y
TRIANGLKINEQUALITY
Iv. wi < IIvllllwll
IIv + wll < IIvll + Ilwll
Example 5 Find cos () for v = [ i ]
and w = [ ; ] and check both inequalities.
Solution The dot product is v· w = 4. Both v and w have length,J5. The cosine is 4/5.
v· w
4 4
cos () = Ilvllllwll - ,J5../5 - "5
The angle is below 900 because v· w = 4 is positive. By the Schwarz inequality, v· w = 4
is less than IIvllllwll = 5. Side 3 = Ilv + wll is less than side 1 + side 2, by the triangle
inequality. For v + w = (3,3) that says .JI8 < ../5 + ../5. Square this to get 18 < 20.
Example 6 The dot product of v = (a, b) and w = (b, a) is 2ab. Both lengths are
J a2 + b2 • The Schwarz inequality in this case says that 2ab < a2 + b2•
This is more famous if we write x = a2 and y = b2 • The "geometric mean" JXY
is not larger than the "arithmetic mean" = average !(x + y).
Geometric < Arithmetic
mean
mean
a2 +b2
ab<---
- 2
becomes
r;;;; x +
y
yxy
< 2 .
Example 5 had a = 2 and b = 1. So x = 4 and y = 1. The geometric mean ,.fXY = 2
is below the arithmetic me~n ~(1 + 4) = 2.5.
\
Notes on Computing
Write the components of vas v(l), . .. , v(N) and similarly for w. In FORTRAN, the sum
v + w requires a loop to add components separately. The dot product also uses a loop to
add the separate v(j)w(j). Here are VPLUSW and VDOTW:
FORTRAN
DO 10 J = 1,N
10 VPLUSW(J) = v(J) + w(J)
DO i0J = i,N
10 VDOTW = VDOTW + V(J) * W(J)
MATLAB and also PYTHON work directly with whole vectors, not their components.
No loop is needed. When v and w have been defined, v + w is immediately understood.

 

Suppose, there are two countries, two products and one input (labor hour). Table 1 shows total output per labor hour. Output Table Mexico Corn 17 bushel Computer 2 1 Suppose the world price for computer (Pw) is 16 (bushel com per computer). Calculate the gains from trade both for USA and Mexico, respectively. Recommended Answer Format (show your calculation); 1. Opportunity costs USA: Corn = Computer- Mexico: Corn= USA 31 bushel Computer 2. Gains from trade USA: Mexico: