Thread: null vector

if a,b,c, are three non co planar vectors such that d.a=d.b=d.c=0(where d.a represent scalar product )then show that d is null vector.

Originally Posted by ayushdadhwal

if a,b,c, are three non co planar vectors such that d.a=d.b=d.c=0(where d.a represent scalar product )then show that d is null vector.

Without additional hypothesis the statement is false!

Consider the orthonormal basis of












Note that

but none of the vectors are zero!

If you are working in three dimensions, then a, b, and c must form a basis (otherwise they would not be non-planar) so d can be written as d= xa+ yb+ zc for numbers x, y, and z. Take the dot product of d with xa+ yb+ zc.