Let the given points be A (−1, −1, 2), B (2, K, 5), C (3, 11, 6). Then
→
AB
=(2+1)
→
i
+(k+1)
→
j
+(5−2)
→
k
=3
→
i
+(k+1)
→
j
+3
→
k
And
→
AC
=(3+1)
→
i
+(3+1)
→
j
+(2−2)
→
k
=4
→
i
+4
→
j
+4
→
k
Now, if A,B and C are collinear, then
→
AB
=λ
→
AC
⇒3
→
i
+(
→
k
+1)
→
j
+3
→
k
=λ(4
→
i
+4
→
j
+4
→
k
) Comparing the coefficient of vector i, we get ⇒ 3 = 4 λ ∴ λ = 3/4 Now, comparing the coefficient of vector j, we get ⇒ (k + 1) = 4 λ ⇒ k + 1 = 4 × (3/4) ⇒ k + 1 = 3 ∴ k = 2