Let O  be the origin. Let OP¯=xi+yj−k , OQ¯=−i+2j+3xk, and OR¯=3i+zj−7k x,y,z∈R,x>0be such that PQ=20,  OP¯  is orthogonal to OQ¯,  P,Q,R  are collinear  then the value of x2+y2+z2

OP⊥OQ¯⇒-x+2y-3x=0⇒y=2x…|PQ¯|=20⇒(x+1)2+(2-y)2+(3x+1)2=20⇒14x2-14=0⇒x=±1,x>0 x=1, y=2OP,OQ,OR are collinear xy−1−123x3z−7=0x−14−3xz−y7−9x−1−z−6=0−14−3z−2−2+z+6=0−14−3z+4+z+6=0−2z=4z=−2Therefore, x2+y2+z2=1+4+4=9