IF `overlinea,overlineb,overlinec` are non-coplanar, then prove that the points with position vectors `2overlinea+3overlineb-overlinec,overlinea-2overlineb+3overlinec,3overlinea+4overlineb-2overlinec,overlinea-6overlineb+6overlinec` are coplanar.

IF overlinea,overlineb,overlinec are noncoplanar, find the point of intersection of the line passing through the points 2overlinea+3overlineb-overlinec,3overlinea+4overlineb-2overlinec with the line joining the points overlinea-2overlineb+3overlinec,overlinea-6overlineb+6overlinec

IF overlinea=overlinei-2overlinej-3overlinek,overlineb=2overlinei+overlinej-overlinek and overlinec=overlinei+3overlinej-2overlinek ,find overlinea.(overlinebtimesoverlinec) .

Show that the four points -overlinea+4overlineb-3overlinec,3overlinea+2overlineb-5overlinec,-3overlinea+8overlineb-5overlinec,-3overlinea+2overlineb+overlinec are coplanar, where overlinea,overlineb,overlinec are non-coplanar vectors.

Fo any four vectors overlinea,overlineb,overlinec and overlined , prove that (i) (overlineatimesoverlineb)times(overlinectimesoverlined)=[overline(ac)overlined]overlineb-[overlineboverlinecoverlined]overlinea and (ii) (overlineatimesoverlineb)times(overlinectimesoverlined)=[overlineaoverline(bd)]overlinec-[overlineaoverlineboverlinec]overlined

Let overlinea=overlinei+2overlinej+3overlinek & overlineb=3overlinei+overlinej . Find a unit vector in the direction of overlinea+overlineb

IF overlinea+overlineb+overlinec=overline0,|overlinea|=3|overlineb|=5 and |overlinec|=7 , then find the between overline a and overlineb .

IF overlinea=2overlinei+overlinej-overlinek,overlineb=-overlinei+2overlinej-4overlinek,overlinec=overlinei+overlinej+overlinek then find (overlinea times overlineb).(overlineb times overlinec)

IF overlinea=2overlinei-overlinek,overlineb=-overlinei+2overlinej-4overlinek,overlinec=overlinei+overlinej+overlinek then find (overlineatimesoverlineb).(overlinebtimesoverlinec)

IF overlinea=overlinei+2overlinej-3overlinek and overlineb=3overlinei-overlinej+2overlinek then show that overlinea+overlineb and overlinea-overlineb are perpendicular to each other.