Class 12

Math

Algebra

Vector Algebra

Find the position vector of the mid point of the vector joining the points P(2, 3, 4) and$Q(4,1,2)$.

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If $a,bandc$ are three non-zero non-coplanar vectors, then find the linear relation between the following four vectors: $a−2b+3c,2a−3b+4c,3a−4b+5c,7a−11b+15⋅$

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

If $a,bandc$ are three non-zero vectors, no two of which ar collinear, $a+2b$ is collinear with $c$ and $b+3c$ is collinear with $a,$ then find the value of $∣∣ a+2b+6c∣∣ ˙$

If $4i^+7j^ +8k^,2i^+3j^ +24and2i^+5j^ +7k^$ are the position vectors of the vertices $A,BandC,$ respectively, of triangle $ABC$ , then the position vecrtor of the point where the bisector of angle $A$ meets $BC$ is a. $32 (−6i^−8j^ −k^)$ b. $32 (6i^+8j^ +6k^)$ c. $31 (6i^+13j^ +18k^)$ d. $31 (5j^ +12k^)$

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

Let $G_{1},G_{2}andG_{3}$ be the centroids of the triangular faces $OBC,OCAandOAB,$ respectively, of a tetrahedron $OABC˙$ If $V_{1}$ denotes the volumes of the tetrahedron $OABCandV_{2}$ that of the parallelepiped with $OG_{1},OG_{2}andOG_{3}$ as three concurrent edges, then prove that $4V_{1}=9V_{1}˙$

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 $a$ ,$b$ ,$c$ ,$d$ are the position vector of point $A,B,C$ and $D$ , respectively referred to the same origin $O$ such that no three of these point are collinear and $a$ + $c$ = $b$ + $d$ , than prove that quadrilateral $ABCD$ is a parallelogram.