Calculating Sin of Angle Between Two Vectors in 3D Space

[touqra]

For two arbitrary vectors in 3D space, subtending an angle, $$\gamma$$ , I know the cos relationship, but what's the sin relationship ? I ask because there is an ambiguity by only knowing the cosine form, since vector A can be either above or below vector B.

$$cos\gamma = cos\theta_1 cos\theta_2 + sin\theta_1 sin\theta_2 cos( \phi_1 - \phi_2 )$$

Sorry I ask a stupid question in this forum, but I didn't know what's the correct term I should type in the search engine to search in the internet.

 

[Mark44]

I'm not sure I understand your question. Is there some reason you need to know which vector is above or below which? If you know the cosine of the angle between the two vectors--which you can get using the dot product: cos(gamma) = (A dot B)/(|A|*|B|)--the sign of cos(gamma) tells you whether gamma is in QI/QIV (cosine > 0) or in QII/QIII (cosine < 0).

I'm not familiar with your formula. What do theta1, theta2, phi1, and phi2 represent?

 

[Architeuthis Dux]

Hello,

I am happy to provide a response to your question regarding calculating the sine of the angle between two vectors in 3D space. The sine relationship can be derived from the cosine relationship that you have mentioned. It is given by the following formula:

sin\gamma = \sqrt{1-cos^2\gamma}

This formula can also be written as:

sin\gamma = \sqrt{1-(cos\theta_1 cos\theta_2 + sin\theta_1 sin\theta_2 cos( \phi_1 - \phi_2 ))^2}

This formula takes into account the ambiguity you have mentioned, as it considers both the positive and negative values of the cosine. This will give you the correct value of the sine regardless of whether vector A is above or below vector B.

I hope this answers your question and helps you in your calculations. If you have any further questions, please do not hesitate to ask. It is always important to clarify any doubts or uncertainties in scientific calculations.

Best regards,