If P⃗× Q⃗=Q⃗× P⃗, the angle between P⃗ and Q⃗ is θ(0° < θ < 360°). The value of 'θ' will be ___________°.

Solution

Physics sir · 4 steps

A cross product changes sign when you swap the order, so this one can only be equal to itself if it is zero.

1. Swap rule. The cross product is anti-commutative: Q⃗×P⃗ = -P⃗×Q⃗. Put this into the given condition.

   P⃗×Q⃗ = -P⃗×Q⃗ ⇒ 2 P⃗×Q⃗ = 0⃗ ⇒ P⃗×Q⃗ = 0⃗

2. Use the magnitude. The magnitude is |P⃗×Q⃗| = PQ sin θ. For non-zero vectors this is zero only when sin θ = 0, that is θ = 0° or 180°.
3. Apply the range. The question allows only 0° < θ < 360°, so θ = 0° is not allowed and 360° is not reached.
4. So. The vectors are antiparallel, and θ = 180°.

The common slip: 0. Taking θ = 0° also makes the cross product zero, but the range 0° < θ < 360° leaves it out, so 0 is not the answer.

The idea to keep. Whenever A⃗×B⃗ equals B⃗×A⃗, the cross product must be zero, so the vectors are parallel or antiparallel. Then let the given range of angles pick the answer.

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