Class 12

Math

Algebra

Vector Algebra

Write all the unit vectors in $XY−plane˙$

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If $a×b=b×c=0,wherea,b,andc$ are coplanar vectors, then for some scalar $k$ prove that $a+c=kb˙$

Show, by vector methods, that the angularbisectors of a triangle are concurrent and find an expression for the position vector of the point of concurrency in terms of the position vectors of the vertices.

For given vector, $a$ = 2$i^$ $j$ +2$k^$ and $b$ = -$i^$ +$j^ $ - $k^$ , find the unit vector in the direction of the vector $a$ +$b$ .

Vectors $a=−4i^+3k^;b=14i^+2j^ −5k^$ are laid off from one point. Vector $d^$ , which is being laid of from the same point dividing the angle between vectors $aandb$ in equal halves and having the magnitude $6 ,$ is a. $i^+j^ +2k^$ b. $i^−j^ +2k^$ c. $i^+j^ −2k^$ d. $2i^−j^ −2k^$

Find the least positive integral value of $x$ for which the angel between vectors $a=xi^−3j^ −k^$ and $b=2xi^+xj^ −k^$ is acute.

If $i^×[(a−j^ )×i^]+j^ ×[(a−k^)×j^ ]+k^×[(a−i^)×k^]=0,$ then find vector $a˙$

$a,b,c$ are three coplanar unit vectors such that $a+b+c=0.$ If three vectors $p ,q ,andr$ are parallel to $a,b,andc,$ respectively, and have integral but different magnitudes, then among the following options, $∣p +q +r∣$ can take a value equal to a. $1$ b. $0$ c. $3 $ d. $2$

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