If given vectors are coplanar, then there exists two scalar quantities x and y such that 2
^
i
−
^
j
+
^
k
= x (
^
i
+2
^
j
−3
^
k
) + y (3
^
i
+a
^
j
+5
^
k
) ... (1) Comparing coefficient of
^
i
,
^
j
and
^
k
on both sides of (1) we get x + 3y = 2 , 2x + ay = –1 , –3x + 5y = 1 ...(2) Solving first and third equations, we get x = 1/2, y = 1/2 Since the vectors are coplanar, therefore these values of x and y will satisfy the equation 2x + ay = –1 ∴ 2 (1/2) + a (1/2) = – 1 ⇒ a = –4