Find a unit vector perpendicular to the plane ABC, where the coordinates of A, B and C are A(3, –1, 2), B(1, –1, –3) and C(4, –3, 1).

Given points A(3, –1, 2), B(1, –1, –3) and C(4, –3, 1)

Let position vectors of the points A, B and C be, and respectively.

We know position vector of a point (x, y, z) is given by, where, and are unit vectors along X, Y and Z directions.

Similarly, we have and

Plane ABC contains the two vectors and.

So, a vector perpendicular to this plane is also perpendicular to both of these vectors.

Recall the vector is given by

Similarly, the vector is given by

We need to find a unit vector perpendicular to and.

Recall a vector that is perpendicular to two vectors and is

Here, we have (a₁, a₂, a₃) = (–2, 0, –5) and (b₁, b₂, b₃) = (1, –2, –1)

Let the unit vector in the direction of be.

We know unit vector in the direction of a vector is given by .

Recall the magnitude of the vector is

Now, we find.

So, we have

Thus, the required unit vector that is perpendicular to plane ABC is.