Class 12

Math

Algebra

Vector Algebra

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.

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$ABCD$ 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 $x_{2}+3y_{2}=3$ be the equation of an ellipse in the $x−y$ plane. $AandB$ are two points whose position vectors are $−3 i^and−3 i^+2k^˙$ 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.

If the vectors $A,B,C$ of a triangle $ABC$ are $(1,2,3),(−1,0,0),(0,1,2),$ respectively then find $∠ABC˙$

In a four-dimensional space where unit vectors along the axes are $i^,j^ ,k^andl^,anda_{1},a_{2},a_{3},a_{4}$ are four non-zero vectors such that no vector can be expressed as a linear combination of others and $(λ−1)(a_{1}−a_{2})+μ(a_{2}+a_{3})+γ(a_{3}+a_{4}−2a_{2})+a_{3}+δa_{4}=0,$ then a. $λ=1$ b. $μ=−2/3$ c. $γ=2/3$ d. $δ=1/3$