Vector Algebra

For real numbers , , , and , consider the matrix M = bmatrix & 1 2 & - 1 2 \ 1 3 & & 1 3 \ & & bmatrix . Suppose that MMT = I , where MT is the transpose of the matrix M, and I is the 3 3 identity matrix. Let u = i + 1 3 j + k , v = 1 2 i + j + k and w = - 1 2 i + 1 3 j + k . Match each entry in List-I to the correct entry in List-II and choose the correct option.

1. (P) → (5), (Q) → (4), (R) → (2), (S) → (1)
2. (P) → (4), (Q) → (5), (R) → (1), (S) → (2)
3. (P) → (5), (Q) → (3), (R) → (2), (S) → (1)
4. (P) → (5), (Q) → (4), (R) → (1), (S) → (2)

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 .