The base of the roof, 𝑂𝐴𝐵𝐶, is rectangular and horizontal with 𝑂𝐴 = 𝐶𝐵 = 14 𝑚 and 𝑂𝐶 = 𝐴𝐵 = 8 𝑚. The top of the roof 𝐷𝐸 is 5 𝑚 above the base and DE = 6m. The sloping edges 𝑂𝐷,𝐶𝐷,𝐴𝐸 and 𝐵𝐸 are all equal in length.

Question

The base of the roof, 𝑂𝐴𝐵𝐶, is rectangular and horizontal with 𝑂𝐴 = 𝐶𝐵 = 14 𝑚 and 𝑂𝐶 = 𝐴𝐵 = 8 𝑚. The top of the roof 𝐷𝐸 is 5 𝑚 above the base and DE = 6m. The sloping edges 𝑂𝐷,𝐶𝐷,𝐴𝐸 and 𝐵𝐸 are all equal in length.
Unit vector i and j are parallel to OA and OC respectively and the unit vectors k is vertically upwards.
(i) Express the vector 𝑂𝐷⃗⃗⃗⃗⃗⃗ in terms of i , j and k, and find its magnitude (4 marks)
(ii) Use a scalar product to find angle 𝐷𝑂𝐵. (5 marks)

oobleck · Answer

D is 4m in from AC, so OD = 4i+4j+4k OB = 14i+8j If θ = ∡DOB, then OB•OD = |OB| * |OD| cosθ I assume you know how to find the dot product and magnitudes