Answer:
[tex]\theta = \frac{\pi}{4}[/tex]
Explanation:
Given
[tex]\uparrow A * B = B * \uparrow A[/tex]
Required
Determine the angle between A and B
We start with:
[tex]\uparrow A * B = ABsin\theta[/tex]
and
[tex]B * \uparrow A = ABcos\theta[/tex]
Subtract both equations
[tex]ABsin\theta - ABcos\theta = \uparrow A B - B\uparrow A[/tex]
[tex]ABsin\theta - ABcos\theta = 0[/tex]
[tex]ABsin\theta = ABcos\theta[/tex]
Divide both sides by AB --- assume no null vectors
[tex]sin\theta = cos\theta[/tex]
Divide both sides by [tex]cos\theta[/tex]
[tex]\frac{sin\theta}{cos\theta} = \frac{cos\theta}{cos\theta}[/tex]
[tex]tan\theta = 1[/tex]
Take tan inverse of both sides
[tex]\theta = tan^{-1}(1)[/tex]
[tex]\theta = 45^\circ[/tex]
Convert to radians
[tex]\theta = \frac{180}{4}^\circ[/tex]
[tex]\theta = \frac{\pi}{4}[/tex]
