Q.Are the four points $A(1,-1,1)$, $B(-1,1,1)$, $C(1,1,1)$ and $D(2,-3,4)$ coplanar? Justify your answer.

Vectors that can be drawn lying in one common plane (or in parallel planes) are called coplanar. Two non-collinear vectors a and b always determine a plane, and a third vector r lies in that same plane exactly when it can be written uniquely as r = t1 a + t2 b for some scalars t1, t2 -- this expression is called a linear combination of a and b. Extending the idea, three vectors a, b, c are coplanar exactly when there is a non-zero combination x a + y b + z c = 0 (with x, y, z not all zero); if a, b, c are NOT coplanar, the only way x a + y b + z c can equal zero is with x=y=z=0, and such vectors are called linearly independent. Four points with position vectors a, b, c, d are coplanar exactly when there are scalars x, y, z, w, not all zero, with x a + y b + z c + w d = 0 and x+y+z+w = 0. Once the vector (cross) and scalar triple products are introduced later in the chapter, coplanarity gets a cleaner, purely computational test: three vectors are coplanar exactly when their scalar triple product [a b c] is zero. …