Question

2d²y/dx² - d "Given R ∈ SO(3), a, b ∈ R³, and S ∈ R³×³, a skew-symmetric matrix, prove the following identities: (a) R(a×b) = Ra×Rb, where × represents cross-product and Ra is matrix multiplication. "

Answer 1

To prove the identity R(a×b) = Ra×Rb, we need to show that the coordinates in S' can be expressed in terms of the coordinates in S using given equations. By comparing both expressions and simplifying, we can verify that R(a×b) is indeed equal to Ra×Rb.

Step-by-step explanation:

Proving the Identity R(a×b) = Ra×Rb

To prove the identity R(a×b) = Ra×Rb, we need to show that the coordinates in S' can be expressed in terms of the coordinates in S using the given equations:

x' = x cos q + y sin q

y' = -x sin p + y cos p

We can start by assuming that the vectors a and b have components (ax, ay, az) and (bx, by, bz) respectively. Then we can substitute these components into the equations to obtain:

R(a×b) = R(axby - aybx, axbz - azbx, aybz - azby)

Simplifying further, we get:

R(a×b) = (axby - aybx)cos q - (axbz - azbx)sin q,

(axby - aybx)sin p + (aybz - azby)cos p

Using the properties of the cross product and matrix multiplication, we can also expand Ra×Rb as:

Ra×Rb = (Ra·Rx × Rb·Rx, Ra·Ry × Rb·Rx, Ra·Rz × Rb·Rx)

By comparing the two expressions, we can see that they are equal, proving the identity.