Class 12

Math

Algebra

Vector Algebra

If either $a=0$and $b=0$then $a×b=0$. Is Is the converse true? Justify your answer with an example.

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

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

Let $a=i−k,b=xi+j +(1−x)k$ and $c=yi+xj +(1+x−y)k$ . Then $[abc]$ depends on (A) only $x$ (B) only $y$ (C) Neither $x$ nor $y$ (D) both $x$ and $y$

If the vectors $3p +q ;5p−3q and2p +q ;4p −2q $ are pairs of mutually perpendicular vectors, then find the angle between vectors $p andq ˙$

If $a=2i^+3j^ −5k^,b=mi^+nj^ +12k^anda×b=0,$ then find $(m,n)˙$

A vector has components $p$ and 1 with respect to a rectangular Cartesian system. The axes are rotted through an angel $α$about the origin the anticlockwise sense. Statement 1: IF the vector has component $p+2$and 1 with respect to the new system, then $p=−1.$ Statement 2: Magnitude of the original vector and new vector remains the same.

Show that the points $A(1,−2,−8),B(5,0,−2)andC(1,3,7)$ are collinear, and find the ratio in which $B$ divides $AC˙$

Statement 1: If $uandv$ are unit vectors inclined at an angle $αandx$ is a unit vector bisecting the angle between them, then $x=(u+v)/(2sin(α/2)˙$ Statement 2: If $DeltaABC$ is an isosceles triangle with $AB=AC=1,$ then the vector representing the bisector of angel $A$ is given by $AD=(AB+AC)/2.$