If `(overlinea,overlineb)=30^@` then `(overlinea,-overlineb)=`

Let overlinea and overlineb be vectors, satisfying |overlinea|=|overlineb|=5 and (overlinea,overlineb)=45^@ . Find the area of the triangle having overlinea-2overlineb and 3overlinea+2overlineb as two of its sides.

If overline a,overline b,overline c are vectors of length 4, 4, 5 re-spectively, and overline a,overline b,overline c are perpendicular to overlineb +overline c,c+overlinea, overlinea + overlineb, then |overlinea+overlineb+overlinec|=

For any three vectors overlinea,overlinebandoverlinec,(overlinea-overlineb).[(overlineb+overlinec)times(overlinec+overlinea)] is equal to :

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

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

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.

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=2overlinei+3overlinej+4overlinek,overlineb=overlinei+overlinej-overlinek,overlinec=overlinei-overlinej+overlinek , compute overlineax(overlinebxoverlinec) and verify that it is perpendicular to overlinea