The midpoint of two opposite sides of a quadrilateral and the midpoint of the diagonals are the vertices of a parallelogram. Prove that using vectors.
ABCD 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 OA+OB+OC+OD=xOE,thenx is equal to a. 3 b. 9 c. 7 d. 4
ABC is a triangle and P any point on BC. if PQ is the sum of AP + PB +PC , show that ABPQ is a parallelogram and Q , therefore , is a fixed point.
If xandy are two non-collinear vectors and a, b, and c represent the sides of a ABC satisfying (a−b)x+(b−c)y+(c−a)(×xy)=0, then ABC is (where ×xy is perpendicular to the plane of xandy ) a. an acute-angled triangle b. an obtuse-angled triangle c. a right-angled triangle d. a scalene triangle
Let x2+3y2=3 be the equation of an ellipse in the x−y plane. AandB are two points whose position vectors are −3i^and−3i^+2k^˙ Then the position vector of a point P on the ellipse such that ∠APB=π/4 is a. ±j^ b. ±(i^+j^) c. ±i^ d. none of these
If a,b,c are non-coplanar vector and λ is a real number, then the vectors a+2b+3c,λb+μcand(2λ−1)c are coplanar when a. μ∈R b. λ=21 c. λ=0 d. no value of λ
A unit vector of modulus 2 is equally inclined to x - and y -axes at an angle π/3 . Find the length of projection of the vector on the z -axis.