Long Answer
Medium difficulty • Structured explanation
Question 1
Long FormDerive the expression for the magnitude and direction of the resultant of two vectors A and B making an angle θ, using the parallelogram law.
- Place vectors A and B with tails at common origin O; complete the parallelogram OQSP. The resultant R = A + B is the diagonal OS.
- Drop perpendicular SN from S to extended OP. Using geometry: ON = A + B cos θ and SN = B sin θ.
- Applying Pythagoras: R² = OS² = ON² + SN² = (A + B cos θ)² + (B sin θ)² = A² + B² + 2AB cos θ.
- Therefore R = √(A² + B² + 2AB cos θ), which is the law of cosines.
- For direction angle α with A: tan α = SN/(OP + PN) = B sin θ/(A + B cos θ).
- From the law of sines: R/sin θ = A/sin β = B/sin α, giving complete information about the resultant's direction.