A point `O` is the centre of a circle circumscribed about a triangle`A B Cdot` Then ` vec O Asin2A+ vec O Bsin2B+ vec O Csin2C` is equal to

vec a .(vec b + vec c) is equal to

Let D ,Ea n dF be the middle points of the sides B C ,C Aa n dA B , respectively of a triangle A B Cdot Then prove that vec A D+ vec B E+ vec C F= vec0 .

A sector O A B O of central angle theta is constructed in a circle with centre O and of radius 6. The radius of the circle that is circumscribed about the triangle O A B , is 6costheta/2 (b) 6sectheta/2 c) 3sectheta/2 (d) 3(costheta/2+2)

A sector O A B O of central angle theta is constructed in a circle with centre O and of radius 6. The radius of the circle that is circumscribed about the triangle O A B , is (a) 6costheta/2 (b) 6sectheta/2 (c) 3sectheta/2 (d) 3(costheta/2+2)

' I ' is the incentre of triangle A B C whose corresponding sides are a , b ,c , rspectively. a vec I A+b vec I B+c vec I C is always equal to a. vec0 b. (a+b+c) vec B C c. ( vec a+ vec b+ vec c) vec A C d. (a+b+c) vec A B

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]) .

A B C D is a quadrilateral. E is the point of intersection of the line joining the midpoints of the opposite sides. If O is any point and vec O A+ vec O B+ vec O C+ vec O D=x vec O E ,t h e nx is equal to a. 3 b. 9 c. 7 d. 4

Prove that the resultant of two forces acting at point O and represented by vec O B and vec O C is given by 2 vec O D ,where D is the midpoint of BC.

If the vector product of a constant vector vec O A with a variable vector vec O B in a fixed plane O A B be a constant vector, then the locus of B is a. a straight line perpendicular to vec O A b. a circle with centre O and radius equal to | vec O A| c. a straight line parallel to vec O A d. none of these

If A B C D is a rhombus whose diagonals cut at the origin O , then proved that vec O A+ vec O B+ vec O C+ vec O D+ vec Odot