Class 12

Math

Algebra

Vector Algebra

Find the vector joining the points $P(2,3,0)$and $Q(1,2,4)$directed from P to Q.

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If vector $a+b$ bisects the angle between $a$ and $b$ , then prove that $∣a∣$ = $∣∣ b∣∣ $ .

If in parallelogram ABCD, diagonal vectors are $AC=2i^+3j^ +4k^$ and $BD=−6i^+7j^ −2k^,$ then find the adjacent side vectors $AB$ and $AD$

Statement 1: The direction cosines of one of the angular bisectors of two intersecting line having direction cosines as $l_{1},m_{1},n_{1}andl_{2},m_{2},n_{2}$ are proportional to $l_{1}+l_{2},m_{1}+m_{2},n_{1}+n_{2}˙$ Statement 2: The angle between the two intersection lines having direction cosines as $l_{1},m_{1},n_{1}andl_{2},m_{2},n_{2}$ is given by $cosθ=l_{1}l_{2}+m_{1}m_{2}+n_{1}n_{2}˙$

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.

$ABCD$ 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 $APOP +BQOQ +CROR +DSOS =1.$

The position vectors of the points $PandQ$ with respect to the origin $O$ are $a=i^+3j^ −2k^$ and $b=3i^−j^ −2k^,$ respectively. If $M$ is a point on $PQ,$ such that $OM$ is the bisector of $∠POQ,$ then $OM$ is a. $2(i^−j^ +k^)$ b. $2i^+j^ −2k^$ c. $2(−i^+j^ −k^)$ d. $2(i^+j^ +k^)$

If $AO+OB=BO+OC$ , then $A,BnadC$ are (where $O$ is the origin) a. coplanar b. collinear c. non-collinear d. none of these

The position vectors of points $AandB$ w.r.t. the origin are $a=i^+3j^ −2k^$, $b=3i^+j^ −2k^$ respectively. Determine vector $OP$ which bisects angle $AOB,$ where $P$ is a point on $AB˙$