Question

Show that if A and B are orthogonal matrices, then A B is an orthogonal matrix.

Answer 1

since A and B are orthogonal matrix,
then A A^T = A^TA = I ,and BB^T = B^TB = I

now, (AB)^T =B^TA^T ,

then (AB)(AB)^T = ABB^TA^T = A I A^T =AA^T = I

similarly, (AB)^T(AB) = B^TA^TAB = B^T I B = B^T B = I

hence , (AB)(AB)^T =(AB)^T(AB) = I , => AB is orthogonal

Answer 2

well you can set this up like this
for 2 lines to be orthogonal they are perpendicular
and the simplest 2 perpendicular lines are x , z= 0 and y, z =0
so the vectors for y,z = 0 is
and for y, z = 0 is < 0, kj, 0>
for 2 vectors to be orthogonal their dot product is 0
and taking the dot product these 2 vectors are indeedorthogonal

the dot product is <0,0,0>
which is also orthogonal to both vectors