Question #224510

Let u=(2,6,-7), v=(-1,-1,8), and 3. If (2, 14, 11)=ku+lv what is the value of l?


1
Expert's answer
2021-08-10T13:06:55-0400

AnswerGiven:u=(2,6,7)v=(1,1,8)ku+lv=(2,14,11)(i)ku=k(2,6,7)=(2k,6k,7k)lv=l(1,1,8)=(l,l,8l)ku+lv=(2kl,6kl,7k+8l)(ii)Comparing (i) and (ii),2kl=26kl=147k+8l=11Hence,2kl2=0(iii)6kl14=0(iv)7k+8l11=0(v)By solving either (iii) and (iv), (iv) and (v),or (iii) and (v) simultaneously, we obtaink=3,l=4.Hence, the value of l=4.Answer\\ \textsf{Given:}\\ u=(2,6,-7)\\ v=(-1,-1,8)\\ ku+lv= (2, 14, 11)----------(i)\\ ku=k(2,6,-7)=(2k,6k,-7k)\\ lv=l(-1,-1,8)=(-l,-l,8l)\\ ku+lv=(2k-l,6k-l,-7k+8l)---(ii)\\ \textsf{Comparing }(i)\textsf{ and }(ii),\\ 2k-l=2\\ 6k-l=14\\ -7k+8l=11\\ \textsf{Hence,}\\ 2k-l-2=0------------(iii)\\ 6k-l-14=0-----------(iv)\\ -7k+8l-11=0----------(v)\\ \textsf{By solving either (iii) and (iv), (iv) and (v),} \\ \textsf{or (iii) and (v) simultaneously, we obtain}\\ k=-3, l=-4.\\ \textsf{Hence, the value of } l=-4.



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