Question

If $\vec{P} \times \vec{Q} = \vec{Q} \times \vec{P}$, the angle between $\vec{P}$ and $\vec{Q}$ is $\theta$ (0°<$\theta$<360°). The value of '$\theta$' will be ________°.

Hint:

The cross product of two vectors is zero if and only if the vectors are parallel or anti-parallel (collinear). The condition $\vec{A} \times \vec{B} = \vec{B} \times \vec{A}$ is equivalent to $\vec{A} \times \vec{B} = \vec{0}$.

Answer 1

Correct Answer: 180

The vector cross product is anti-commutative by definition. This means that for any two vectors \( \vec{P} \) and \( \vec{Q} \): 
\( \vec{P} \times \vec{Q} = -(\vec{Q} \times \vec{P}) \). 
The problem states that \( \vec{P} \times \vec{Q} = \vec{Q} \times \vec{P} \).  
We can substitute the anti-commutative property into the given equation: 
\( \vec{P} \times \vec{Q} = -(\vec{P} \times \vec{Q}) \). 
Rearranging the terms, we get: 
\( 2(\vec{P} \times \vec{Q}) = \vec{0} \). 
This implies that the cross product of \( \vec{P} \) and \( \vec{Q} \) must be the zero vector: 
\( \vec{P} \times \vec{Q} = \vec{0} \). 
The magnitude of the cross product is given by \( |\vec{P} \times \vec{Q}| = |\vec{P}||\vec{Q}|\sin\theta \), where \( \theta \) is the angle between the vectors. 
For the cross product to be zero, assuming \( \vec{P} \) and \( \vec{Q} \) are non-zero vectors, we must have \( \sin\theta = 0 \). 
The angles for which \( \sin\theta = 0 \) in the range \( 0^\circ \le \theta<360^\circ \) are \( \theta = 0^\circ \) and \( \theta = 180^\circ \). 
The problem specifies the range as \( 0^\circ<\theta<360^\circ \). 
Therefore, the only possible value for \( \theta \) is \( 180^\circ \).