Question

Consider a pyramid OPQRS located in the first octant (x ≥ 0, y ≥ 0, z ≥ 0) with O as origin, and OP and OR along the x-axis and the y-axis, respectively. The bases OPQR of the pyramid is a square with OP = 3. The point S is directly above the mid-point T of the diagonal OQ such that TS = 3. Then

(a) the acute angle between OQ and OS is π/3

(b) the equation of the plane containing ΔOQS is x – y = 0

(c) the length of the perpendicular from P to the ΔOQS is 3/√2

(d) the perpendicular distance from O to the straight line containing RS is √(15/2)  

Answer 1

Answer is (d) the perpendicular distance from O to the straight line containing RS is √(15/2)

Points O, P, Q, R, S are (0, 0, 0), (3, 0, 0), (3, 3, 0), (0, 3, 0), (3/2, 3/2, 0) respectively

The angle between OQ nad OS is cos^–1(1/√3).

The equation of the plane containing the points O, Q and S is x – y = 0

The perpendicular distance from P(3, 0, 0) to the plane x – y = 0 is

|(3 – 0)/√2| = 3/√2 

and the perpendicular distance from O(0, 0, 0) to the line RS