`A_1,A_2,..., A_n` are the vertices of a regular plane polygon with n sides and O as its centre. Show that `sum_(i=1)^n vec (OA)_i xx vec(OA)_(i+1)=(1-n)(vec (OA)_2 xx vec(OA)_1)`

A_(1),A_(2), …. A_(n) are the vertices of a regular plane polygon with n sides and O ars its centre. Show that sum_(i=1)^(n-1) (vec(OA_(i))xxvec(OA)_(i+1))=(1-n) (vec(OA)_2 xx vec(OA)_(1))

Let A_1,A_2,....A_n be the vertices of an n-sided regular polygon such that , 1/(A_1A_2)=1/(A_1A_3)+1/(A_1A_4) . Find the value of n.

ABCD is a quadrilateral and E is the point of intersection of the lines joining the middle points of opposite side. Show that the resultant of vec (OA) , vec(OB) , vec(OC) and vec(OD) = 4 vec(OE) , where O is any point.

If vec r_1, vec r_2, vec r_3 are the position vectors of the collinear points and scalar p a n d q exist such that vec r_3=p vec r_1+q vec r_2, then show that p+q=1.

In a four-dimensional space where unit vectors along the axes are hat i , hat j , hat ka n d hat l ,a n d vec a_1, vec a_2, vec a_3, vec a_4 are four non-zero vectors such that no vector can be expressed as a linear combination of others and (lambda-1)( vec a_1- vec a_2)+mu( vec a_2+ vec a_3)+gamma( vec a_3+ vec a_4-2 vec a_2)+ vec a_3+delta vec a_4=0, then a. lambda=1 b. mu=-2//3 c. gamma=2//3 d. delta=1//3

On the xy plane where O is the origin, given points, A(1,0), B(0,1) and C(1,1). Let P,Q, and R be moving points on the line OA, OB, OC respectively such that vec(OP)=45t(vec(OA)), vec(OQ)=60t(vec(OB)), vec(OR)=(1-t)(vec(OC)) with t gt 0 . If the three points P,Q and R are collinear then the value of t is equal to

Orthocenter of an equilateral triangle ABC is the origin O. If vec(OA)=veca, vec(OB)=vecb, vec(OC)=vecc , then vec(AB)+2vec(BC)+3vec(CA)=

If vec e_1, vec e_2, vec e_3a n d vec E_1, vec E_2, vec E_3 are two sets of vectors such that vec e_idot vec E_j=1,ifi=ja n d vec e_idot vec E_j=0a n difi!=j , then prove that [ vec e_1 vec e_2 vec e_3][ vec E_1 vec E_2 vec E_3]=1.

vec p , vec q ,a n d vec r are three mutually perpendicular vectors of the same magnitude. If vector vec x satisfies the equation vec pxx(( vec x- vec q)xx vec p)+ vec qxx(( vec x- vec r)xx vec q)+ vec rxx(( vec x- vec p)xx vec r)=0,t h e n vec x is given by a. 1/2( vec p+ vec q-2 vec r) b. 1/2( vec p+ vec q+ vec r)"" c. 1/3( vec p+ vec q+ vec r) d. 1/3(2 vec p+ vec q- vec r)

If a_1,a_2,a_3,... are in A.P. and a_i>0 for each i, then sum_(i=1)^n n/(a_(i+1)^(2/3)+a_(i+1)^(1/3)a_i^(1/3)+a_i^(2/3)) is equal to