If a=3ˆi+4ˆj+5ˆkandb=2ˆi+ˆj4ˆk ;How will you find out the component of a perpendicular to b?

1 Answer
Jun 3, 2016

aT=121{83ˆi,94ˆj,65ˆk}

Explanation:

Given two non null vectors a and b the first a allways can be decomposed as a sum of two components: one parallel to b and another perpendicular to b.

The parallel component is the projection of a onto b or
aP=a,bbbb=a,bbb2
and the perpendicular component given by
aT=aaP=aa,bbb2

So index P for parallel and T for perpendicular. We can verify that

aP+aT=a
aP,aT=a,bbb2,aa,bbb2=0
aT,b=aa,bbb2,b=0

In our case

aP=1021{2ˆi,ˆj,4ˆk}
aT=121{83ˆi,94ˆj,65ˆk}