The base of the roof, 𝑂𝐴𝐡𝐢, is rectangular and horizontal with 𝑂𝐴 = 𝐢𝐡 = 14 π‘š and 𝑂𝐢 = 𝐴𝐡 = 8 π‘š. The top of the roof 𝐷𝐸 is 5 π‘š above the base and DE = 6m. The sloping edges 𝑂𝐷,𝐢𝐷,𝐴𝐸 and 𝐡𝐸 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 𝐷𝑂𝐡. (5 marks)

1 answer

D is 4m in from AC, so
OD = 4i+4j+4k

OB = 14i+8j
If θ = ∑DOB, then
OBβ€’OD = |OB| * |OD| cosΞΈ
I assume you know how to find the dot product and magnitudes