If`" "veca=3hati+4hatj+5hatk and vec b= 2hati+hatj-4hatk` ;How will you find out the component of `" "veca " ""perpendicular to" " " vecb`?

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)`

The component of `veca` perpendicular to `vecb`

`=veca-"projection of "veca" " on " " vecb`

`=veca-(veca*vecb)/abs(vecb)xxvecb/abs(vecb)`

`=veca-(veca*vecb)/abs(vecb)^2xxvecb`

`=(3hati+4hatj+5hatk)-((3hati+4hatj+5hatk).(2hati+hatj-4hatk))/abs(2hati+hatj-4hatk)^2xx(2hati+hatj-4hatk)`

`=(3hati+4hatj+5hatk)-(2*3+4*1+5*-4)/(2^2+1^2+4^2)xx(2hati+hatj-4hatk)`

`=(3hati+4hatj+5hatk)+10/21xx(2hati+hatj-4hatk)`

`=1/21[21(3hati+4hatj+5hatk)+10(2hati+hatj-4hatk)]`

`=1/21[83hati+94hatj+65hatk]`