Let #vec a = vec a_1 + vec a_2# with #vec a_1 ne vec 0, vec a_2 ne vec 0#such that #<< vec a_1, vec a_2 >> = 0# and # vec a_2 = lambda vec b# with #lambda in RR#
Making now #<< vec a, vec b >> = << vec a_1+vec a_2 , vec b>> = << vec a_1, vec b >> + << vec a_2, vec b >># but by hypothesis # vec a_2 = lambda vec b# so
#<< vec a, vec b >> = lambda << vec b, vec b >># so
#lambda = (<< vec a, vec b >>)/(<< vec b, vec b >>)# so
#vec a_2 = (<< vec a, vec b >>)/(<< vec b, vec b >>) vec b# and
#vec a_1 = vec a - (<< vec a, vec b >>)/(<< vec b, vec b >>) vec b#
or
#vec a_1 = (3,4,5)-(3 xx 2+4 xx 1-5 xx 4)/(2^2+1^2+4^2)(2,1,-4) = (83/21, 94/21, 65/21)#