The correct option is
D 3^i+2^j+^kPosition vector of A is ^i+^j+^k, gives coordinates as (1,1,1).
Position vector of B is 4^i+^j+^k, gives coordinates as (4,1,1).
Position vector of C is 4^i+5^j+^k, gives coordinates as (4,5,1).
→AB=→B−→A=3^i
→BC=→C−→B=4^j
→AC=→C−→A=3^i+4^j
Here, we can see that △ABC is a right angled triangle with right angle at B.
|→AB|=√32=3
|→BC|=√42=4
|→AC|=√32+42=5
Point of Intersection of angle bisectors can be found using formula:
I=|→BC|×→A+|→CA|×→B+|→AB|×→C|→AB|+|→BC|+|→CA|
I=4×(^i+^j+^k)+5×(4^i+^j+^k)+3×(4^i+5^j+^k)3+4+5
⇒I=112(36^i+24^j+12^j)=3^i+2^j+^k
Hence, incenter will be 3^i+2^j+^k