class 12

Math

Algebra

Vector Algebra

Let $a,b,andc$ be three non coplanar unit vectors such that the angle between every pair of them is $3π $. If $a×b+b×x=pa+qb+rc$ where p,q,r are scalars then the value of $q_{2}p_{2}+2q_{2}+r_{2} $ is

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If $a,b,andc$ be three non-coplanar vector and $a_{prime},b_{prime}andc_{′}$ constitute the reciprocal system of vectors, then prove that $r=(ra˙_{′})a+(rb˙_{_{′}})b+(rc˙_{_{′}})c$ $r=(ra˙_{′})a_{′}+(rb˙_{_{′}})b_{′}+(rc˙_{′})c_{′}$

A ship is sailing towards the north at a speed of 1.25 m/s. The current is taking it towards the east at the rate of 1 m/s and a sailor is climbing a vertical pole on the ship at the rate of 0.5 m/s. Find the velocity of the sailor in space.

If $A,B,C,D$ are four distinct point in space such that $AB$ is not perpendicular to $CD$ and satisfies $ABC˙D=k(∣∣ AD∣∣ _{2}+∣∣ BC∣∣ _{2}−∣∣ AC∣∣ _{2}−∣∣ BD∣∣ _{2}),$ then find the value of $k˙$

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

Let $ABCD$ be a p[arallelogram whose diagonals intersect at $P$ and let $O$ be the origin. Then prove that $OA+OB+OC+OD=4OP˙$

$P(1,0,−1),Q(2,0,−3),R(−1,2,0)andS(,−2,−1),$ then find the projection length of $PQonRS˙$

Statement 1: If $uandv$ are unit vectors inclined at an angle $αandx$ is a unit vector bisecting the angle between them, then $x=(u+v)/(2sin(α/2)˙$ Statement 2: If $DeltaABC$ is an isosceles triangle with $AB=AC=1,$ then the vector representing the bisector of angel $A$ is given by $AD=(AB+AC)/2.$

$_{′}I_{′}$ is the incentre of triangle $ABC$ whose corresponding sides are $a,b,c,$ rspectively. $aIA+bIB+cIC$ is always equal to a. $0$ b. $(a+b+c)BC$ c. $(a+b+c)AC$ d. $(a+b+c)AB$