If A and B are orthogonal, what is norm of A cross B?

The norm of the cross product of two vectors A and B is equal to the product of the norms of the two vectors multiplied by the sine of the angle between them.

Since A and B are orthogonal, the angle between them is 90 degrees, and the sine of 90 degrees is 1. Therefore, the norm of A cross B is equal to the product of the norms of A and B.

||A X B|| = ||A|| - ||B|| - sin(theta)

Since A and B are orthogonal, theta = 90 degrees. Therefore, sin(theta) = 1.

||A X B|| = ||A|| - ||B|| - 1 = ||A|| - ||B||

1

If vectors 𝐀 and 𝐁 are orthogonal, the norm of their cross product 𝐀×𝐁 can be calculated using the formula:


    𝐀×𝐁 = 𝐀𝐁sin(θ)


where θ is the angle between the two vectors. For orthogonal vectors, θ = 90^∘ (or π/2 radians), and thus sin(90^∘) = 1.

Therefore, the norm of the cross product becomes:


    𝐀×𝐁 = 𝐀𝐁


In conclusion, if 𝐀 and 𝐁 are orthogonal, the norm of their cross product is simply the product of their magnitudes:


    𝐀×𝐁 = 𝐀𝐁


Source from chatGPT4