Three vectors OP⃗,OQ⃗ and OR⃗ each of magnitude A are acting as shown in figure. The resultant of the three vectors is A√x. The value of x is _________.

Solution

Physics sir · 4 steps

Add OP and OQ first: their resultant is exactly perpendicular to OR, so Pythagoras finishes it.

1. Pair OP and OQ. OP⃗ and OQ⃗ are equal and at 90°. Their resultant has magnitude √(A² + A²) = A√2 and points along the bisector, 45° above OP.
2. Angle with OR. OR⃗ is 45° below OP. So the angle between A√2 and OR⃗ is 45° + 45° = 90°.

   R = √((A√2)² + A²) = √(2A² + A²) = A√3

3. Check by components. Along OP: A + A/√2. Along OQ: A - A/√2. Squaring and adding gives A²(1 + 1/2 + 1 + 1/2) = 3A², the same result.
4. So. The resultant is A√3, so x = 3.

The common slip: 1.73. Reporting the value of √3 ≈ 1.73 gives the ratio R/A, not x. The question asks for the number under the root, which is 3.

The idea to keep. With three or more vectors, combine the easy pair first. If that partial resultant comes out perpendicular to the remaining vector, Pythagoras gives the answer in one line.

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