Class 12

Math

Algebra

Vector Algebra

Column I, Column II Collinear vectors, p.$a$ Coinitial vectors, q. $b$ Equal vectors, r. $c$ Unlike vectors (same intitial point), s. $d$

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$ABCD$ is a tetrahedron and $O$ is any point. If the lines joining $O$ to the vrticfes meet the opposite faces at $P,Q,RandS,$ prove that $APOP +BQOQ +CROR +DSOS =1.$

If the resultant of two forces is equal in magnitude to one of the components and perpendicular to it direction, find the other components using the vector method.

Statement 1: $∣a∣=3,∣∣ b∣∣ =4and∣∣ a+b∣∣ =5,then∣∣ a−b∣∣ =5.$ Statement 2: The length of the diagonals of a rectangle is the same.

Given three vectors $a=6i^−3j^ ,b=2i^−6j^ andc=−2i^+21j^ $ such that $α=a+b+c$ Then the resolution of the vector $α$ into components with respect to $aandb$ is given by a. $3a−2b$ b. $3b−2a$ c. $2a−3b$ d. $a−2b$

$A,B,C,D$ are any four points, prove that $ABC˙D+BCA˙D+CAB˙D=0.$

Let $a=i−k,b=xi+j +(1−x)k$ and $c=yi+xj +(1+x−y)k$ . Then $[abc]$ depends on (A) only $x$ (B) only $y$ (C) Neither $x$ nor $y$ (D) both $x$ and $y$

A vector has components $p$ and 1 with respect to a rectangular Cartesian system. The axes are rotted through an angel $α$about the origin the anticlockwise sense. Statement 1: IF the vector has component $p+2$and 1 with respect to the new system, then $p=−1.$ Statement 2: Magnitude of the original vector and new vector remains the same.

Check whether the three vectors $2i^+2j^ +3k^,−3i^+3j^ +2k^and3i^+4k^$ from a triangle or not