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Show that the lines whose direction cosines are given by l+m+n=0 and 2mn+3ln-5lm=0 are perpendicular to each other
Condition : if two lines are perpendicular to each other is :-l1.l2+m1.m2+n1.n2=0. [where l1,m1,n1 are the direction cosine of 1st line andl2,m2 ,n2 are the direction cosine of 2nd line.]l+m+n=0…………….(1). ,thus , l1=l , m1=m and n1=n.2mn+3nl+(-5lm)=0……..(2). thus , l2=2mn , m2= 3nl and n2= -5lmNow , l1.l2+m1.m2+n1.n2=l.2mn+m.3nl+n.(-5lm)=5lmn-5lmn =0.=> Thus , the two lines are perpendicular to each other. Proved.
Condition : if two lines are perpendicular to each other is :-
l1.l2+m1.m2+n1.n2=0. [where l1,m1,n1 are the direction cosine of 1st line and
l2,m2 ,n2 are the direction cosine of 2nd line.]
l+m+n=0…………….(1). ,thus , l1=l , m1=m and n1=n.
2mn+3nl+(-5lm)=0……..(2). thus , l2=2mn , m2= 3nl and n2= -5lm
Now , l1.l2+m1.m2+n1.n2=l.2mn+m.3nl+n.(-5lm)=5lmn-5lmn =0.
=> Thus , the two lines are perpendicular to each other. Proved.
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