Class 12

Math

Algebra

Vector Algebra

Find the angle between two vectors $→a$and $→b$with magnitudes 1 and 2 respectively and when $→a→˙b=1$.

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

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

If $i^−3j^ +5k^$ bisects the angle between $a^and−i^+2j^ +2k^,wherea^$ is a unit vector, then a. $a^=1051 (41i^+88j^ −40k^)$ b. $a^=1051 (41i^+88j^ +40k^)$ c. $a^=1051 (−41i^+88j^ −40k^)$ d. $a^=1051 (41i^−88j^ −40k^)$

$a,b,c$ 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. $2$

If $A,B,C,D$ are four distinct point in space such that $AB$ is not perpendicular to $CD$ and satisfies $ABC˙D=k(∣∣ AD∣∣ _{2}+∣∣ BC∣∣ _{2}−∣∣ AC∣∣ _{2}−∣∣ BD∣∣ _{2}),$ then find the value of $k˙$

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.

If the angel between unit vectors $aandb60_{0}$ , then find the value of $∣∣ a−b∣∣ ˙$

The position vectors of points $AandB$ w.r.t. the origin are $a=i^+3j^ −2k^$, $b=3i^+j^ −2k^$ respectively. Determine vector $OP$ which bisects angle $AOB,$ where $P$ is a point on $AB˙$