CAT4MBA

a, b, c, u, v and w are real numbers and a^2 + b^2 + c^2 = 25, u^2 + v^2 + w^2 = 36 and au + bv + cw = 30

Q1. What is the value of (a + b + c)/(u + v + w) ?
a. 25/36                 b. 5/6                   c. 125/216                      d. ¾

Solution:

Note that (a^2 + b^2 + c^2)(u^2 + v^2 + w^2) = (au + bv + cw)

So, according to Cauchy Schwarz Inequality, this equality sign holds whenever there exists a constant 's' such that:
a = ku, b = kv, c = kw

Substituting this in equation (1), we get:

a^2 + b^2 + c^2 = 25

or, (k^2)(u^2 + v^2 + w^2) = 25

or, k^2 = 25/36

or, k = 5/6

So, a/u = b/v = c/w = k = (a + b + c)/(u + v + w) = 5/6

Thank You

Ravi Raja

a, b, c, u, v and w are real numbers and a^2 + b^2 + c^2 = 25, u^2 + v^2 + w^2 = 36 and au + bv + cw = 30

Q2. What is the value of  (a + b)/ (u+v)? (a, b, u, v ≠ 0)
a. 3/4                 b. 5/6                    c. 25/36              d. cannot be determined

Solution:

Note that (a^2 + b^2 + c^2)(u^2 + v^2 + w^2) = (au + bv + cw)

So, according to Cauchy Schwarz Inequality, this equality sign holds whenever there exists a constant 's' such that:
a = ku, b = kv, c = kw

Substituting this in equation (1), we get:

a^2 + b^2 + c^2 = 25

or, (k^2)(u^2 + v^2 + w^2) = 25

or, k^2 = 25/36

or, k = 5/6

So, a/u = b/v = k = (a + b)/(u + v) = 5/6

Thank You

Ravi Raja

Note that (a^2 + b^2 + c^2)(u^2 + v^2 + w^2) = (au + bv + cw)

So, according to Cauchy Schwarz Inequality, this equality sign holds whenever there exists a constant 's' such that:
a = ku, b = kv, c = kw

Substituting this in equation (1), we get:

a^2 + b^2 + c^2 = 25

or, (k^2)(u^2 + v^2 + w^2) = 25

or, k^2 = 25/36

or, k = 5/6

So, a/u = c/w = k

2a/2u = 3c/3w = k = (2a + 3c)/(2u + 3w) = 5/6

Thank You

Ravi Raja

a, b, c, u, v and w are real numbers and a^2 + b^2 + c^2 = 25, u^2 + v^2 + w^2 = 36
a=b=0 and c =5
u=0=v and w=6

(a + b + c )/(u +v +w) = 5/6

Can some one confirm me if it is correct or not