Given, a, b and c be the distinct non-negative numbers. The vectors a
^
i
+a
^
j
+c
^
k
,
^
i
+
^
k
,c
^
i
+c
^
j
+b
^
k
lie on a plane So the vectors a
^
i
+a
^
j
+c
^
k
,
^
i
+
^
k
,c
^
i
+c
^
j
+b
^
k
are coplaner. ⇒|
a
a
c
1
0
1
c
c
b
|=0 ⇒a(0−c)−a(b−c)+c(c−0)=0 ⇒−ac−ab+ac+c2=0 ⇒c2=ab So, c is the geometric mean of a and b.Hence, Let a, b and c be the distinct non-negative numbers. If the vectors a
^
i
+a
^
j
+c
^
k
,
^
i
+
^
k
,c
^
i
+c
^
j
+b
^
k
lie on a plane, then the statement "c is the geometric mean of a and b. " is correct.