Class 12

Math

Algebra

Vector Algebra

If the vectors $3p +q ;5p−3q and2p +q ;4p −2q $ are pairs of mutually perpendicular vectors, then find the angle between vectors $p andq ˙$

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In a quadrilateral $PQRS,PQ=a,Q R,b,SP=a−b,M$ is the midpoint of $Q RandX$ is a point on $SM$ such that $SX=54 SM˙$ Prove that $P,XandR$ are collinear.

If $∣∣ a+b∣∣ <∣∣ a−b∣∣ ,$ then the angle between $aandb$ can lie in the interval a. $(π/2,π/2)$ b. $(0,π)$ c. $(π/2,3π/2)$ d. $(0,2π)$

If the resultant of three forces $F_{1}=pi^+3j^ −k^,F_{2}=6i^−k^andF_{3}=−5i^+j^ +2k^$ acting on a parricle has magnitude equal to 5 units, then the value of $p$ is a. $−6$ b. $−4$ c. $2$ d. $4$

Two forces $AB$ and $AD$ are acting at vertex A of a quadrilateral ABCD and two forces $CB$ and $CD$ at C prove that their resultant is given by 4$EF$ , where E and F are the midpoints of AC and BD, respectively.

$A,B,CandD$ have position vectors $a,b,candd,$ respectively, such that $a−b=2(d−c)˙$ Then a. $ABandCD$ bisect each other b. $BDandAC$ bisect each other c. $ABandCD$ trisect each other d. $BDandAC$ trisect each other

Prove that the sum of three vectors determined by the medians of a triangle directed from the vertices is zero.

Let $u^=i^+j^ ,v^=i^−j^ andw^=i^+2j^ +3k^˙$ If $n^$ is a unit vector such that $u^n^˙=0andv^n^˙=0,$ then find the value of $∣∣ w^n^˙∣∣ ˙$

In a triangle $ABC,DandE$ are points on $BCandAC,$ respectivley, such that $BD=2DCandAE=3EC˙$ Let $P$ be the point of intersection of $ADandBE˙$ Find $BP/PE$ using the vector method.