The correct option is D −6√21
Given, →a=2^i−2^j+^k,→b=^i+2^j−2^k and →c=2^i−^j+4^k
We need to find the projection of (→a+2→b) on →c
We know that the projection of (→a+2→b) on →c is given by (→a+2→b).^c, where ^c is the unit vector of →c.
So, we have
^c=→c|→c|=2^i−^j+4^k√22+(−1)2+42=1√21(2^i−^j+4^k)
Also, (→a+2→b)=(2^i−2^j+^k)+2(^i+2^j−2^k)
⇒ (→a+2→b)=4^i+2^j−3^k
Hence, projection is given by
(→a+2→b).^c=(4^i+2^j−3^k).(1√21(2^i−^j+4^k))
⇒ 8−2−12√21=−6√21