Given
-
#vecp=hati+2hatj-2hatk#
-
#vecq=2hati-hatj+2hatk#
we are to find #vecm and vecn# satisfying conditions
By condition (b)#vecn# is parallel to #vec p#
So
- We can write
#vecn=alpha*vecp#,where #alpha# is a constant.
#vecn=alphavecp=alphahati+2alphahatj-2alphahatk#
By condition (c)
#vecm+vecn=vecq#
#=>vecm=vecq-vecn#
#=>vecm=(2-alpha)hati-(1+2alpha)hatj+(2+2alpha)hatk#
Now by condition (a) #vecm# is perpendicular to #vec p#
So #vecm*vecp=0#
#=>1xx(2-alpha)-2(1+2alpha)-2(2+2alpha)=0#
#=>2-alpha-2-4alpha-4-4alpha=0#
#=>alpha=-4/9#
Now
#vecn=alphahati+2alphahatj-2alphahatk#
#=>vecn=-4/9hati-8/9hatj+8/9hatk#
and
#vecm=(2-alpha)hati-(1+2alpha)hatj+(2+2alpha)hatk#
#=>vecm=(2+4/9)hati-(1-8/9)hatj+(2-8/9)hatk#
#=>vecm=22/9hati-1/9hatj+10/9hatk#
