7 Let u and y be vectors in a real inner Product space V and

7. Let u and y be vectors in a real inner Product space V and suppose that ||u|| = 3, ||v|| = 5, and (u, v) = -2. (i) Find (2u - v, u + 3v). (ii) Find ||2u - 3v||.

Solution

a) <(2u-v),(u+3v)>

=> 2 (u.u) + 6 (u.v) - (v.u) - 3(v.v)

=> 2||u||^2 + 5*(u.v) - 3||v||^2

=> 2 * (3)^2 + 5 * (-2) - 3*(5)^2

=> 18 - 10 - 75

=> -67

b) <(2u-3v),(2u-3v)>

=> 4 (u.u) - 6* (u.v) - 6* (u.v) + 9||v||^2

=> 4 * 9 - 12*2 + 9 * (5)^2

=> 36 - 24 + 175

=> 187

Hence ||2u - 3v|| = (187)^(1/2) units