Vector Algebra

For any two points M and N in the XY-plane, let MN denote the vector from M to N, and 0 denote the zero vector. Let P, Q and R be three distinct points in the XY-plane. Let S be a point inside the triangle PQR such that SP + 5 SQ + 6 SR = 0. Let E and F be the mid-points of the sides PR and QR, respectively. Then the value of length of the line segment EFlength of the line segment ES is .

Consider the vectors x= i+2 j+3 k, y=2 i+3 j+ k, z=3 i+ j+2 k. For two distinct positive real numbers and , define X= x+ y- z, Y= y+ z- x, Z= z+ x- y. If the vectors X, Y, and Z lie in a plane, then the value of + -3 is .

Let OP= -1 i+ j+ k, OQ= i+ -1 j+ k and OR= i+ j+ 12 k be three vectors, where , R-\0\ and O denotes the origin. If ( OP OQ ) OR=0 and the point ( , ,2) lies on the plane 3x+3y-z+l=0, then the value of l is .

Let p=2 i+ j+3 k and q= i- j+ k. If for some real numbers , , and , we have 15 i+10 j+6 k= (2 p+ q )+ ( p-2 q )+ ( p q ), then the value of is .