Two forces whose resultant is 100N are at right angle to eachother. if one of them makes an angle of 30° with the resultant, determine its magnitude​

Let F₁ and F₂ denote the two forces, and R the resultant force.

F₁ and F₂ point perpendicularly to one another, so their dot product is

F₁ • F₂ = 0

We're given that one of these vectors, say F₁, makes an angle with R of 30°, so that

F₁ • R = ||F₁|| ||R|| cos(30°)

But we also have

F₁ • R = F₁ • (F₁ + F₂) = (F₁ • F₁) + (F₁ • F₂) = F₁ • F₁ = ||F₁||²

So, knowing that ||R|| = 100 N, we get that

(100 N) ||F₁|| cos(30°) = ||F₁||²

(100 N) cos(30°) = ||F₁||

||F₁|| ≈ 86.6 N

(And the same would be true for F₂.)