Show that `[[ax-by-cz, ay+bx, az+cx],[bx+ay, by-cz-ax, bz+cy], [cx+az, cy+bz, cz-ax-by]]=(a^2+b^2+c^2)(ax+by+cz)(x^2+y^2+z^2)`

Prove the following: |[ax-by-cz,ay+bx,az+cx],[bx+ay,by-cz-ax,bz+cy],[cx+az,ay+bz,cz-ax-by]| = (a^2+b^2+c^2)(ax+by+cz)(x^2+y^2+z^2)

det[[ Prove that ax-by-cz,ay+bx,cx+azay+bx,by-cz-ax,bz+cycx+az,bz+cy,cz-ax-by]]=(x^(2)+y^(2)+z^(2))(a^(2)+b^(2)+c^(2))(ax+by+cz)

Let a,b,c be real numbers with a^2 + b^2 + c^2 =1. Show that the equation |[ax-by-c,bx+ay,cx+a],[bx+ay,-ax+by-c,cy+b],[cx+a,cy+b,-ax-by+c]|=0 represents a straight line.

Let a,b,c be real numbers with a^2 + b^2 + c^2 =1 . Show that the equation |[ax-by-c,bx+ay,cx+a],[bx+ay,-ax+by-c,cy+b],[cx+a,cy+b,-ax-by+c]|=0 represents a straight line.

If x = cy + bz, y = az + cx , z =bx + ay, then prove that x^2/(1-a^2)= y^2/(1-b^2)

If x = cy + bz, y = az + cx,z = bx + ay where x,y,z are not all zero, prove that a^2 + b^2 + c^2 + 2ab = 1.

STATEMENT-1 : If a, b, c are distinct and x, y, z are not all zero and ax + by + cz = 0, bx + cy + az = 0, cx + ay + bz = 0 , then a + b + c = 0 and STATEMENT-2 : a^2 + b^2 + c^2 > ab + bc + ca , if a, b, c are distinct.

If ay - bx/c = cx - az/b = bz - cy/a, then prove that x/a = y/b = z/c.