Class 12

Math

Algebra

Vector Algebra

For any two vectors $a$ and $b$, we always have $∣∣ a+b∣∣ ≤∣a∣+∣∣ b∣∣ $

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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

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

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 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.

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.

$ABCD$ parallelogram, and $A_{1}andB_{1}$ are the midpoints of sides $BCandCD,$ respectivley . If $∀_{1}+AB_{1}=λAC,thenλ$ is equal to a. $21 $ b. $1$ c. $23 $ d. $2$ e. $32 $

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

Show that the point $A,B$ and $C$ with position vectors $a$ =3$i^$ - 4$j^ $ -4$k^$ = 2$i^$ $j$ + $k^$ and $c$ = $i^$ - 3$j^ $ - 5$k^$ , respectively from the vertices of a right angled triangle.