Using vector method, show that the points A(1, -1, 0), B(4, -3, 1) and C(2, -4, 5) are the vertices of a right-angled triangle.

If the position vectors of the vertices A, B and C of a ∆ABC be and respectively, prove that ∆ABC is equilateral.

Show that the points A, B and C having position vectors and respectively, form the vertices of a right-angled triangle.

Find the position vector of the point which divides the join of the points and (i) internally and (ii) externally in the ratio 2 : 3.

The position vectors of two points A and B are and respectively. Find the position vector of a point C which divides AB externally in the ratio 1 : 2. Also, show that A is the mid-point of the line segment CB.