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 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 adot vec b)+m n( vec bdot vec c)+n l( vec cdot 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 vec a,vec b and vec c are any three non-coplanar vectors,then prove that points l_(1)vec a+m_(1)vec b+n_(1)vec c,l_(2)vec a+m_(2)vec b+n_(2)vec c,l_(3)vec a+m_(3)vec b+n_(3)vec c,l_(4)vec a+m_(4)vec b+n_(4)vec c are coplanar if [[n_(1),n_(2),n_(3),n_(4)1,1,1,1]]=0

If |(a,b,c),(l,m,n),(p,q,r)|=2 , then what is the value of the determinant |(6a,3b,15c),(2l,m,5n),(2p,q,5r)| = ?

If the points ((a^(3))/(a-1),(a^(2)-3)/(a-1)),((b^(3))/(b-1),(b^(3)-3)/(b-1)) and ((c^(3))/(c-1),(c^(3)-3)/(c-1)) where a,b,c are different from 1 lie on the line lx+my+n=0a+b+c=(m)/(l)ab+bc+ca+(n)/(l)=0abc=((3m+n))/(l)abc-(bc+ca+ab)+3(a+b+c)=0

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

Let a, b, c be the side-lengths of a triangle, and l, m,n be the lengths of its medians. Put K=((l+m+n)/(a+b+c)) Then, as a, b, c vary, K can assume every value in the interval

If y=det[[f(x),g(x),h(x)l,m,na,b,c]] prove that ,f'(x),g'(x),h'(x)l,m,na,b,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