`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)`

C is the midpoint of the line segment bar(AB) and O is any point outside AB, show that vec(OA)+vec(OB)=2vec(OC) .

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.

Show that ("lim")_(n vec oo)(1/(n+1)+1/(n+2)++1/(6n))=log6

ABCD is a quadrilateral and E is the point of intersection of the lines joining the mid-points of opposite sides. If O be any point in the plane, then show that vec(OA)+vec(OB)+vec(OC)+vec(OD)=4vec(OE) .

Let vec(a),vec(b),vec (c ) be the positions vectors of the vertices of a triangle , prove that the area of the triangle is 1/2| vec(a) xx vec(b) + vec(b) xx vec(c)+ vec(c)xx vec(a)|

Given four points P_1,P_2,P_3a n dP_4 on the coordinate plane with origin O which satisfy the condition ( vec (O P))_(n-1)+( vec(O P))_(n+1)=3/2 vec (O P)_(n) . If P1 and P2 lie on the curve xy=1 , then prove that P3 does not lie on the curve

The intercept made by the plane vec r . vec n=q on the x-axis is a. q/( hat i dot vec n) b. ( hat i dot vec n)/q c. ( hat i dot vec n)/q d. q/(| vec n|)

In a four-dimensional space where unit vectors along the axes are hat i , hat j , hat k and hat l , and 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

If a_1,a_2,.....a_n are positive real number whose product is a fixed number c, then the minimum value of a_1+a_2+......+a_(n-1)+a_n is

If a_1,a_2,.....a_n are in H.P., then the expression a_1a_2 + a_2a_3 + ... + a_(n-1)a_n is equal to