Three Dimensional Geometry

Let L1 be the line of intersection of the planes given by the equations 2x + 3y + z = 4 and x + 2y + z = 5. Let L2 be the line passing through the point P(2, -1, 3) and parallel to L1. Let M denote the plane given by the equation 2x + y - 2z = 6. Suppose that the line L2 meets the plane M at the point Q. Let R be the foot of the perpendicular drawn from P to the plane M. Then which of the following statements is (are) TRUE?

1. The length of the line segment PQ is 9 3
2. The length of the line segment QR is 15
3. The area of PQR is 32 234
4. The acute angle between the line segments PQ and PR is -1 ( 12 3 )

Let R be such that the lines L1: x+111= y+212= z+293 and L2: x+163= y+112= z+4 intersect. Let R1 be the point of intersection of L1 and L2. Let O=(0,0,0), and n denote a unit normal vector to the plane containing both the lines L1 and L2. Match each entry in List-I to the correct entry in List-II.

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

A straight line drawn from the point P(1,3,2), parallel to the line x-21= y-42= z-61, intersects the plane L1: x-y+3z=6 at the point Q. Another straight line which passes through Q and is perpendicular to the plane L1 intersects the plane L2: 2x-y+z=-4 at the point R. Then which of the following statements is (are) TRUE?

1. The length of the line segment PQ is 6
2. The coordinates of R are (1,6,3)
3. The centroid of the triangle PQR is ( 43, 143, 53 )
4. The perimeter of the triangle PQR is 2+ 6+ 11