The magnitude of vectors OA⃗, OB⃗ and OC⃗ in the given figure are equal. The direction of OA⃗ + OB⃗ - OC⃗ with x-axis will be :

Solution

Physics sir · 3 steps

Break all three into x and y parts, flip the signs of OC⃗, then take y over x.

1. Read the angles. Let each vector have magnitude a. OA⃗ is 30° above the +x axis, and OB⃗ is 60° below the +x axis. OC⃗ is 45° above the −x axis, so its x-part is negative.
2. Subtracting OC. Subtracting OC⃗ is the same as adding -OC⃗. So flip the sign of both its components before adding.

   Vector	x-component (in units of a)	y-component (in units of a)
   OA⃗	cos 30° = √3/2	sin 30° = 1/2
   OB⃗	cos 60° = 1/2	-sin 60° = -√3/2
   -OC⃗	+1/√2	-1/√2
   Sum	(1+√3+√2)/2	(1-√3-√2)/2

   tan θ = R_y/Rₓ = (1-√3-√2)/(1+√3+√2)

3. So. Here 1/√2 = √2/2, so every component has the same factor a/2, and it cancels in the ratio. The x-part is positive and the y-part is negative, so the resultant points about 27° below the +x axis. The answer is (a).

The idea to keep. Resolve every vector into components first, and a minus sign just flips both components. Then tan θ = R_y/Rₓ, and the signs of Rₓ and R_y tell you the quadrant.

Answer from the NTA final answer key. Solution by PrepWiser.