If `|overlinea|=3,|overlineb|=4,|overlinea-overlineb|=5` then `|overlinea+overlineb|=....`

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|=

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

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.

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

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)

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.