Let →a=4^i+5^j−^k, →b=^i−4^j+5^k and →c=3^i+^j−^k. Find a vector →d which is perpendicular to both →c and →b and →d⋅→a=21.

Given: →a=4^i+5^j−^k, →b=^i−4^j+5^k, →c=3^i+^j−^k
Let →d=x^i+y^j+z^k
Also given: →d⋅→c=0 (1)
→d⋅→b=0 (2)
and →d⋅→a=21 (3)
Now, in (1): (x^i+y^j+z^k)⋅(3^i+^j−^k)=0
⇒3x+y−z=0
Similarly from (2) and (3):
x−4y+5z=0
and 4x+5y−z=21
Solving these equations in x, y, z we get:
x=15, y=163, z=133
Thus, the vector →d=x^i+y^j+z^k can be written as :
→d=15^i+163^j+133^k