Coplanarity is one of the most tested concepts in JEE vectors. Three vectors a, b, c are coplanar if and only if [a b c]=0, meaning one is a linear combination of the other two.

Test for three vectors: compute the 3x3 determinant of their components. If zero, they're coplanar.

Test for four points: Given A, B, C, D, compute [AB AC AD]. If zero, the four points lie in a common plane. The choice of base vertex doesn't matter — [AB AC AD]=0 iff [BA BC BD]=0.

Test for two lines: Lines r=a1+tb1 and r=a2+sb2 are coplanar iff [a2-a1, b1, b2]=0. If additionally b1 x b2 != 0, the lines intersect; if b1 x b2 = 0, they're parallel.

Linear dependence: In 3D, three vectors are linearly dependent iff they're coplanar iff [a b c]=0. Any four or more vectors in 3D are always linearly dependent.

Common problem types: (1) Find the value of a parameter for which given vectors are coplanar — set the determinant to zero and solve. (2) Prove that given vectors/points are coplanar — show the determinant equals zero. (3) Find the equation of a plane through three points using the coplanarity condition for a general point.

The coplanarity condition [a b c]=0 is equivalent to saying the parallelepiped has zero volume — it has been "flattened" into a plane. This geometric interpretation helps in visualizing the condition.