Suppose V1 v2 v3 are distinct points on one line in Rpi The

Suppose V_1, v_2, v_3 are distinct points on one line in R^pi. The line need not pass through the origin. Prove that {v_1, v_2, v_3} is a linearly dependent set.

Solution

Since the points lie on the same line we must have

v1-v2=k(v2-v3) where k is osme real number

ie v1-v2 and v2-v3 give the same parallel vector differing only by a factor of a constant

So,

v1-v2-kv2+kv3=0

v1+(-1-k)v2+kv3=0

Hence the set is linearly dependent