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Consider a pyramid OPQRS located in the ...

Consider a pyramid OPQRS located in the first octant `(xge0, yge0, zge0)` with O as origin and OP and OR along the X-axis and the Y-axis , respectively. The base OPQRS of the pyramid is a square with OP=3. The point S is directly above the mid point T of diagonal OQ such that TS=3. Then,

A

` pi/3`

B

` pi/6`

C

` cos^(-1) "" 1/sqrt3`

D

` cos^(-1)"" 1/3`

Text Solution

Verified by Experts

The correct Answer is:
C
The coordinates of T and S are ( 3/2,3/2,0) and ( 3/2, 3/2,3) respectively.
` OvecQ= 3 hati + 3hatj and OvecS = 3/2 hati + 3/2 hatj +3 hatk`

Let ` theta` the angle between ` OvecQ and OvecS`.then
` cos theta = ( OvecQ. OvecS)/(|OvecQ||OvecS|`
`Rightarrow cos theta= ( 9/2 +9/2+0)/(sqrt(9 +9) sqrt( 9/4 +9/4+9))`
` Rightarrow cos theta 9/(3sqrt2 xx sqrt(27/2))= 1/sqrt3`
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