If p+q+r=0=a+b+c, then write the value of the determinant `|[pa,qb,rc],[qc,ra,pb],[rb,pc,qa]|`

Evaluate the following determinants: [[a-b,b-c,c-a],[x-y,y-z,z-x],[p-q,q-r,r-p]]

If vec(a) .vec(b) = 0 , then write the value of vec(a) xx(vec(b) xx vec(c))

If A, B, P, Q and R are the five points in a plane, then show that the sum of the vectors vec(AP), vec(AQ), vec(AR), vec(PB), vec(QB) and vec(RB) is 3 vec(AB) .

An expression of the form (a+b+c+d+ .... ) consisting of sum of many distinct symbols is called a multinomial. Show that (a+b+c)^n is the sum of all terms of the form n!/p!q!e!a^pb^qc^r where p, q and r range over all possible triples of non negative integers such that p+q+r = n.

Prove that the following. [[a,b,c],[x,y,z],[p,q,r]]=[[y,b,q],[x,a,p],[z,c,r]]=[[x,y,z],[p,q,r],[a,b,c]]

A variable plane meets the coordinate axes at P, Q, R points. If the plane passes through a fixed point (a, b, c), prove that the centre of the shpere passing the origin and P, Q, R will lie on the surface a/x+b/y+c/z=2

If a,b,c are respectively the sums of p,q,r terms of an A.P., prove that a/p(q-r)+b/q(r-p)+c/r(p-q)=0