Class 12

Math

Algebra

Vector Algebra

If $a,b,c$are mutually perpendicular vectors of equal magnitudes, show that the vector $a+b+c$is equally inclined to $a,b$ and $c$.

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

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

If the vectors $i^−j^ ,j^ +k^anda$ form a triangle, then $a$ may be a. $−i^−k^$ b. $i^−2j^ −k^$ c. $2i^+j^ +k^$ d. $i^+k^$

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

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 the resultant of two forces is equal in magnitude to one of the components and perpendicular to it direction, find the other components using the vector method.

If $a,b,andc$ are three non-coplanar non-zero vecrtors, then prove that $(a.a)b×c+(a.b)c×a+(a.c)a×b=[bca]a$