Answer:
a) the centre is at (12.5 m, 40 m , 100 m ) with respect to our position
b) the length of the strings S will be 216.85 m
c) the angle that is formed by the strings is 1.23 rad
Step-by-step explanation:
assuming that we stand on one of the corners on the floor , so our coordinates are (0,0,0) , then the coordinates of the center of the gymnasium are found through
x center = (25 + 0)/2 = 12.5 m
y center = (80+ 0)/2 = 40 m
z center = (200+ 0)/2 = 100 m
then the centre is at (12.5 m, 40 m , 100 m ) with respect to our position
b) the length of the strings S will be the modulus of the vector that points from our position to the diagonally opposite corners
|S| = √(25²+80²+200²) = 216.85 m
c) the angle can be found through the dot product of the vectors that represent the strings S₁ and S₂
S₁ =(25,80,10)
S₂ =(-25,80,100)
then
S₁*S₂ = 25*(-25) +80*80 + 100*100 = 15775
but also
S₁*S₂ = |S₁||S₂| cos θ = |S|² * cos θ
S₁*S₂ = |S|² * cos θ
cos θ= S₁*S₂/|S|²
θ= cos ⁻¹ ( S₁*S₂/|S|² ) = cos ⁻¹ [15775/(25²+80²+200²)] = 1.23 rad
