If ` vec a , vec ba n d vec c` are the position vectors of the vertices `A ,Ba n dC` respectively, of ` A B C ,` prove that the perpendicular distance of the vertex `A` from the base `B C` of the triangle `A B C` is `(| vec axx vec b+ vec bxx vec c+ vec cxx vec a|)/(| vec c- vec b|)dot`

Let vec a , vec b ,a n d vec c be any three vectors, then prove that [ vec axx vec b vec bxx vec c vec cxx vec a]=[ vec a vec b vec c]^2dot

If O A B C is a tetrahedron where O is the orogin anf A ,B ,a n dC are the other three vertices with position vectors, vec a , vec b ,a n d vec c respectively, then prove that the centre of the sphere circumscribing the tetrahedron is given by position vector (a^2( vec bxx vec c)+b^2( vec cxx vec a)+c^2( vec axx vec b))/(2[ vec a vec b vec c]) .

Prove that if the vectors vec a, vec b, vec c satisfy vec a+ vec b + vec c = vec 0 , then vec bxx vec c = vec c xx vec a = vec a xx vec b

If [ vec a vec b vec c]=2, then find the value of [( vec a+2 vec b- vec c)( vec a- vec b)( vec a- vec b- vec c)]dot

If vec a, vec b, vec c, vec d respectively are the position vectors representing the vertices A, B, C, D of a parallelogram then write vec d in terms of vec a, vec b and vec c

Prove that [ vec a, vec b, vec c + vec d] = [ vec a, vec b, vec c] + [ vec a, vec b , vec d] .

For any three vectors vec a, vec b , vec c , show that vec a xx (vec b + vec c) + vec b xx (vec c + vec a) + vec c xx (vec a + vec b) = 0

If the vectors vec a , vec b ,a n d vec c form the sides B C ,C Aa n dA B , respectively, of triangle A B C ,t h e n (a) vec a . vec b+ vec b . vec c+ vec c . vec a=0 (b) vec axx vec b= vec bxx vec c= vec cxx vec a (c) vec adot vec b= vec bdot vec c= vec c dot vec a (d) vec axx vec b+ vec bxx vec c+ vec cxx vec a=0

For any four vectors, prove that ( vec bxx vec c)dot( vec axx vec d)+( vec cxx vec a)dot( vec bxx vec d)+( vec axx vec b)dot( vec cxx vec d)=0.

For any three vectors veca, vec b, vec c prove that (vec a + vec b)+ vec c = vec a + (vec b + vec c)