Class 12

Math

Algebra

Vector Algebra

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 $y$

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If the vectors $α=ai^+aj^ +ck^,β =i^+k^andγ =ci^+cj^ +bk^$ are coplanar, then prove that $c$ is the geometric mean of $aandb˙$

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

The position vectors of $PandQ$ are 5$i^+4j^ +ak^$ and $−i^+2j^ −2k^$ , respectively. If the distance between them is 7, then find the value of $a˙$

The vector $a$ has the components $2p$ and 1 w.r.t. a rectangular Cartesian system. This system is rotated through a certain angel about the origin in the counterclockwise sense. If, with respect to a new system, $a$ has components $(p+1)and1$ , then $p$ is equal to a. $−4$ b. $−1/3$ c. $1$ d. $2$

i. Prove that the points $a−2b+3c,2a+3b−4cand−7b+10c$ are are collinear, where $a,b,c$ are non-coplanar. ii. Prove that the points $A(1,2,3),B(3,4,7),andC(−3,−2,−5)$ are collinear. find the ratio in which point C divides AB.

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.

Prove that the necessary and sufficient condition for any four points in three-dimensional space to be coplanar is that there exists a liner relation connecting their position vectors such that the algebraic sum of the coefficients (not all zero) in it is zero.

Let $ABC$ be triangle, the position vecrtors of whose vertices are respectively $i^+2j^ +4k^$ , -2$i^+2j^ +k^and2i^+4j^ −3k^$ . Then $DeltaABC$ is a. isosceles b. equilateral c. right angled d. none of these