`A(veca),B(vecb),C(vecc)` are the vertices of the triangle ABC and `R(vecr)` is any point in the plane of triangle ABC , then `vec r.(vecaxxvecb+vecbxxvecc+veccxxveca)` is always equal to

If A(1, 4, 2), B(-2, 1, 2) and C(2, -3, 4) are the vertices of the triangle ABC, then find the angles of the triangle ABC.

If A (veca).B(vecb) and C(vecc) are three non-collinear point and origin does not lie in the plane of the points A, B and C, then for any point P(vecP) in the plane of the triangleABC such that vector vec(OP) is bot to plane of trianglABC , show that vec(OP)=([vecavecbvecc] (vecaxxvecb+vecbxxvecc+veccxxveca))/(4Delta^(2))

If veca , vecb , vecc are position vectors of the vertices A, B, C of a triangle ABC, show that the area of the triangle ABC is 1/2 ​ [ veca × vecb + vecb × vecc + vecc × veca ] . Also deduce the condition for collinearity of the points A, B and C.

If the points A(z),B(-z),C(1-z) are the vertices of an equilateral triangle ABC then Re(z) is

A(2, 3, 1), B(-2, 2, 0) and C(0, 1, -1) are the vertices of the triangle ABC. Show that the triangle ABC is right -angled.

If vecb and vecc are unit vectors, then for any arbitary vector veca, (((veca xx vecb) + (veca xx vecc))xx (vecb xx vecc)) .(vecb -vecc) is always equal to

D, E, F are the midpoints of the sides bar(BC), bar(CA) " and " bar(AB) respectively of the triangle ABC. If P is any point in the plane of the triangle, show that vec(PA)+vec(PB)+vec(PC)=vec(PD)+vec(PE)+vec(PF) .

If vecA+vecB+vecC=0 then show that vecAxxvecB=vecBxxvecC=vecCxxvecA .

If in a triangle ABC, b sin B = c sin C, then the triangle is-

In triangle ABC, If cos B= a/(2c) , then the triangle is-