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For any three vectors overset(to)(a),...

For any three vectors `overset(to)(a), overset(to)(b) " and " overset(to)(C )`
` (overset(to)(a) - overset(to)(b)). {(overset(to)(b)-overset(to)(c))xx(overset(to)(c)-overset(to)(a))} = 2overset(to)(a).(overset(to)(b)xx overset(to)(c))`

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For any three vectors overset(to)(a) , overset(to)(b) and overset(to) ( c ) , prove that vectors overset(to)(a) - overset(to)(b) , overset(to)(b) - overset(to) ( c), overset(to)(c) - overset(to)(a) are coplanar.

If the vectors overset(to)(b), overset(to)(c ) , overset(to)(d) are not coplanar then prove than the vectors (overset(to)(a) xx overset(to)(b)) xx (overset(to)(c ) xx overset(to)(d)) + (overset(to)(a) xx overset(to)(c )) xx (overset(to)(d) xx overset(to)(b)) +(overset(to)(a) xx overset(to)(d)) xx (overset(to)(b) xx overset(to)( c)) is parallel to overset(to)(a)

if overset(to)(a), overset(to)(b) " and " overset(to)(c ) are unit vectors satisfying |overset(to)(a)-overset(to)(b)|^(2)+|overset(to)(b)-overset(to)(c)|^(2)+|overset(to)(c)-overset(to)(a)|^(2)=9 |2overset(to)(a) +5overset(to)(b)+5overset(to)(c)| is equal to

if overset(to)(a), overset(to)(b) " and " overset(to)(c ) are unit vectors satisfying |overset(to)(a)-overset(to)(b)|^(2)+|overset(to)(b)-overset(to)(c)|^(2)+|overset(to)(c)-overset(to)(a)|^(2)=9 |2overset(to)(a) +5overset(to)(b)+5overset(to)(c)| is equal to

If overset(to)(A), overset(to)(B), overset(to)(C ) three non-coplanar vectors then (overset(to)(A) ,(overset(to)(B)xxoverset(to)(C)))/((overset(to)(C)xx overset(to)(A)). overset(to)(B))+ (overset(to)(B).(overset(to)(A) xx overset(to)(C)))/(overset(to)(C).(overset(to)(A)xx overset(to)(B)))=.........

If overset(to)(A), overset(to)(B), overset(to)(C ) three non-coplanar vectors then (overset(to)(A) ,(overset(to)(B)xxoverset(to)(C)))/((overset(to)(C)xx overset(to)(A)). overset(to)(B))+ (overset(to)(B).(overset(to)(A) xx overset(to)(C)))/(overset(to)(C).(overset(to)(A)xx overset(to)(B)))=.........

if overset(to)(b) " and " overset(to)(c ) are any two non- collinear unit vectors and overset(to)(a) is any vector then (overset(to)(a).overset(to)(b))overset(to)(b).(overset(to)(a).overset(to)(c )) overset(to)(c ) + .(overset(to)(a).(overset(to)(b)xxoverset(to)(c)))/(|overset(to)(b)xxoverset(to)(c)|^(2)).(overset(to)(b)xxoverset(to)(c))=.........

if overset(to)(b) " and " overset(to)(c ) are any two non- collinear unit vectors and overset(to)(a) is any vector then (overset(to)(a).overset(to)(b))overset(to)(b).(overset(to)(a).overset(to)(c )) overset(to)(c ) + .(overset(to)(a).(overset(to)(b)xxoverset(to)(c)))/(|overset(to)(b)xxoverset(to)(c)|^(2)).(overset(to)(b)xxoverset(to)(c))=.........

A,B,C and d are four points in a plane with position vectors overset(to)(a),overset(to)(b),overset(to)(c ) and overset(to)(d) respectively such that (overset(to)(a)-overset(to)(d)).(overset(to)(b)-overset(to)(c))=(overset(to)(b)-overset(to)(d))(overset(to)(c)-overset(to)(a))=0 The point D then is the .... of the DeltaABC

A,B,C and d are four points in a plane with position vectors overset(to)(a),overset(to)(b),overset(to)(c ) and overset(to)(d) respectively such that (overset(to)(a)-overset(to)(d)).(overset(to)(b)-overset(to)(c))=(overset(to)(b)-overset(to)(d))(overset(to)(c)-overset(to)(a))=0 The point D then is the .... of the DeltaABC