If a,b,c,l and m are vectors, prove that
[a b c] `(lxxm)=|(a,b,c),(a*l,b*l,c*l),(a*m,b*m,c*m)|`

If y=|(f(x),g(x),h(x)),(l, m, n),( a, b, c)| , prove that (dy)/(dx)=|(fprime(x),gprime(x),hprime(x)),(l, m, n),( a, b, c)|

If l ,\ m ,\ n are scalars and vec a ,\ vec b ,\ vec c are vectors, prove that |l vec a+m vec b+n vec c|^2=l^2| vec a|^2+m^2| vec b|^2+n^2 | vec c|^2+2\ {l m( vec a . vec b)+m n( vec b . vec c)+n l( vec c . vec a)} . Also deduce that |l vec a+m vec b+n vec c|^2=l^2 |vec a|^2+m^2| vec b|^2+n^2| vec c|^2if\ vec a ,\ vec b ,\ vec c are mutually perpendicular vectors.

If the determinant |(b -c,c -a,a -b),(b' -c',c' -a',a' -b'),(b'' -c'',c'' -a'',a'' - b'')| is expressible an m |(a,b,c),(a',b',c'),(a'',b'',c'')| , then the value of m, is

Write a vector normal to the plane vec r=l vec b+m vec c

If a,b and c are three mutually perpendicular vectors, then the projection of the vectors l(a)/(|a|)+m(b)/(|b|)+n((axxb))/(|axxb|) along the angle bisector of the vectors a and b is

three number a,b,c are in GP such that : (i) a+b+c=70 (ii) 4a,5b,4c are in AP if G=max{a,b,c} and L=min{a,b,c} than [G/L] is equal is :

Let R be a relation on NxxN defined by (a , b)\ R(c , d)hArra+d=b+c\ if\ (a , b),(c , d) in NxxN then show that: (i)\ (a , b)R\ (a , b)\ for\ a l l\ (a , b) in NxxN (ii)\ (a , b)R(c , d)=>(c , d)R(a , b)for\ a l l\ (a , b),\ (c , d) in NxxN (iii)\ (a , b)R\ (c , d)a n d\ (c , d)R(e ,f)=>(a , b)R(e ,f)\ for all \ (a , b),\ (c , d),\ (e ,f) in NxxN

If A=|{:(a,d,l),(b,e,m),(c,f,n):}|" and B = "|{:(l,m,n),(a,b,c),(d,e,f):}| , then

Supose directioncoisnes of two lines are given by u l+vm+wn=0 and al^2+bm^2+cn^2=0 where u,v,w,a,b,c are arbitrary constnts and l,m,n are directioncosines of the lines. For u=v=w=1 if (n_1 n_2)/(l_1 l_2)=((a+b)/(b+c)) then (A) (m_1m_2)/(l_1 l_2)=((b+c))/((c+a)) (B) (m_1m_2)/(l_1 l_2)=((c+a))/((b+c)) (C) (m_1m_2)/(l_1 l_2)=((a+b))/((c+a)) (D) (m_1m_2)/(l_1 l_2)=((c+a))/((a+b))

If a=x^(m+n)\ y^l,\ b=x^(n+l)\ y^m and c=x^(l+m)y^n , prove that a^(m-n)\ b^(n-l)\ c^(l-m\ )=1