POMOGITE RESHIT POJALUYSTA ZADANIE PO ALGEBRE NA ANGLIYSKOM YAZIKE! SPASIBO

Question

SUPPOSE THAT THE DOT PRODUCT OF U =[U1,U2] AND V=[V1,V2] WHERE DEFINED 
AS U*V =5U1V1+2U2V2. WHICH (IF ANY) OF THE FOLLOWING STATEMENTS WOULD BE
 TRUE FOR ALL VECTORS U,V,W AND ALL SCALARS C ?
a. U * V = V * U 
b. U *(V + W) = U * V + U * W
c. (CU) * V = C (U * V) 
d. U * U >= 0 and U * U = 0 if and only if U = 0

Littleking · Accepted Answer

Ответ: a, b, c, d, то есть все.

Александр Холодилин · Answer

Где ж ты раньше был? ))
г) U*U=5*U1U1+2U2U2=5U1²+2*U2²>=0 because U1²>=0 and U2²>=0. U*U=0 if U1²=0 and U2²=0 simultaneously => U1=0 and U2=0 <=> U=0.
vice versa: let U=0, i.e. U=(0;0), U1=U2=0, then U*U=5*0*0+2*0*0=0.