Class 12

Math

Algebra

Vector Algebra

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

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The median AD of the triangle ABC is bisected at E and BE meets AC at F. Find AF:FC.

Prove that the sum of three vectors determined by the medians of a triangle directed from the vertices is zero.

Let $a,b,andcanda_{_{′}},b_{_{′}},c_{′}$ are reciprocal system of vectors, then prove that $a_{_{′}}×b_{_{′}}+b_{_{′}}×c_{_{′}}+c_{_{′}}×a_{_{′}}=[abc]a+b+c $ .

If $ABCD$ is a rhombus whose diagonals cut at the origin $O,$ then proved that $OA+OB+OC+OD+O˙$

The position vector of the points $PandQ$ are $5i^+7j^ −2k^$ and $−3i^+3j^ +6k^$ , respectively. Vector $A=3i^−j^ +k^$ passes through point $P$ and vector $B=−3i^+2j^ +4k^$ passes through point $Q$ . A third vector $2i^+7j^ −5k^$ intersects vectors $AandB˙$ Find the position vectors of points of intersection.

$ABCD$ is a tetrahedron and $O$ is any point. If the lines joining $O$ to the vrticfes meet the opposite faces at $P,Q,RandS,$ prove that $APOP +BQOQ +CROR +DSOS =1.$

In triangle $ABC,∠A=30_{0},H$ is the orthocenter and $D$ is the midpoint of $BC$. Segment $HD$ is produced to $T$ such that $HD=DT$ The length $AT$ is equal to (a). $2BC$ (b). $3BC$ (c). $24 BC$ (d). none of these

If the vectors $c,a=xi^+yj^ +zk^andb=j^ $ are such that $a,candb$ form a right-handed system, then find $⋅$