Class 12

Math

Algebra

Vector Algebra

Consider two points P and Q with position vectors $→OP=3→a−2→b$and $→OQ=→a+→b$Find the position vector of a point R which divides the line joining P and Q in the ratio 2:1, (i) internally, and (ii) externally.

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If $a$ and $b$ are non-collinear vectors and \displaystyle\vec{{A}}={\left({p}+{4}{q}\right)}\vec{{a}}={\left({2}{p}+{q}+{1}\right)}\vec{{b}}{a}{n}{d}\vec{{B}}={\left(-{2}{p}+{q}+{2}\right)}\vec{{a}}+{\left({2}{p}-{3}{q}-{1}\right)}\vec{{b}} ,a n d if$3A=2B$ , then determine p and q.

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 $∣∣ AC∣∣ ˙$

If $∣a∣=5,∣∣ a−b∣∣ =8and∣∣ a+b∣∣ =10$ , then find $∣∣ b∣∣ ˙$

If $D,EandF$ are three points on the sides $BC,CAandAB,$ respectively, of a triangle $ABC$ such that the $CDBD ,AECE ,BFAF =−1$

The lines joining the vertices of a tetrahedron to the centroids of opposite faces are concurrent.

If $a,b,candd$ are four vectors in three-dimensional space with the same initial point and such that $3a−2b+c−2d=0$ , show that terminals $A,B,CandD$ of these vectors are coplanar. Find the point at which $ACandBD$ meet. Find the ratio in which $P$ divides $ACandBD˙$

If $∣a∣=2∣∣ b∣∣ =5$ and $∣∣ a×b∣∣ =8,$ then find the value of $a.b˙$

$ABCD$ is a parallelogram. If $LandM$ are the mid-points of $BCandDC$ respectively, then express $ALandAM$ in terms of $ABandAD$ . Also, prove that $AL+AM=23 AC˙$