"Suppose the relationship ProjaU = ProjaV is true for some vectors a, u and w

Question

 Need help with entire question

 

i) Verify that u.v = v.a

ii) Provide a counter example to show that u need not be equal to v

Answer 1

If u and v are vectors in Rⁿ, then

 

u·v = (u₁v₁ + u₂v₂ + u₃v₃ + ... + u_nv_n) = (v₁u₁ + v₂u₂ + v₃u₃ + ... + v_nu_n) = v·u

 

Example:

 

u = (1, 2)

v = (3, 4)

 

u, v in R²

u ≠ v

 

u·v = (1)(3) + (2)(4) = 3 + 8 = 11

v·u = (3)(1) + (4)(2) = 3 + 8 = 11