Class 12

Math

Algebra

Vector Algebra

Find the direction cosines of the vector joining the points $A(1,2,3)$and$B(1,2,1)$, directed from A to B.

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Let $OACB$ be a parallelogram with $O$ at the origin and$OC$ a diagonal. Let $D$ be the midpoint of $OA˙$ using vector methods prove that $BDandCO$ intersect in the same ratio. Determine this ratio.

Given three points are $A(−3,−2,0),B(3,−3,1)andC(5,0,2)˙$ Then find a vector having the same direction as that of $AB$ and magnitude equal to $∣∣ AC∣∣ ˙$

Statement 1: if three points $P,QandR$ have position vectors $a,b,andc$ , respectively, and $2a+3b−5c=0,$ then the points $P,Q,andR$ must be collinear. Statement 2: If for three points $A,B,andC,AB=λAC,$ then points $A,B,andC$ must be collinear.

If $a,bandc$ are non-coplanar vectors, prove that the four points $2a+3b−c,a−2b+3c,3a+$ 4$b−2canda−6b+6c$ are coplanar.

If 2$AC$ = 3$CB$ , then prove that 2$OA$ =3$CB$ then prove that 2$OA$ + 3$OB$ =5$OC$ where $O$ is the origin.

Show that $(a−b)×(a+b)=2a×b$ and given a geometrical interpretation of it.

ABC is a triangle and P any point on BC. if $PQ$ is the sum of $AP$ + $PB$ +$PC$ , show that ABPQ is a parallelogram and Q , therefore , is a fixed point.

The position vectors of the point $A,B,CandDare3i^−2j^ −k^,2i^+3j^ −4k^,−i^+j^ +2k^$ and 4$i^+5j^ +λk^,$ respectively. If the points $A,B,CandD$ lie on a plane, find the value of $λ˙$