Vectors and orthonormal matrix

Question

For 2(a)(i), are the length of a =Sqrt(14) and b = sqrt(38)?

For (ii), is the angle = 4.31?

For (iii), is the answer 3.73?

For (iv), is the answer -i+j+k?

For (b) to (e), I have no idea what I should do. Would anyone mind advising me?

Answer 1

Hints:

$$||u||=||v||\implies (u+v)\cdot(u-v)=||u||^2-u\cdot v+u\cdot v= -||v||^2=\;\ldots$$

$$P^t=P^*=P^{-1}\implies ||Pu||^2=Pu\cdot Pu=u\cdot P^*Pu=u\cdot Iu=u\cdot u=||u||^2\;,\;\;etc.$$