Bisector of Angle between Two Vectors

Q. The vector c directed along the internal bisector of the angle between the vectors $a=7\hat{i}-4\hat{j}-4\hat{k}andb=-2\hat{i}-\hat{j}+2\hat{k}$ with $|c|=5\sqrt{6}$, is

1. $\frac{5}{3}(\hat{i}-7\hat{j}+2\hat{k})$
2. $\frac{5}{3}(-5\hat{i}+5\hat{j}+2\hat{k})$
3. $\frac{5}{3}(5\hat{i}+5\hat{j}+2\hat{k})$
4. $\frac{5}{3}(\hat{i}+7\hat{j}+2\hat{k})$

Q. The vector$\overrightarrow{V}$ directed along the internal bisector of the angle between the vectors
$\overrightarrow{A}=3\hat{i}-4\hat{j},\overrightarrow{B}=4\hat{i}+2\hat{j}-4\hat{k}$ and $\left(\overrightarrow{V}\right|=2$ is

1. $\frac{38\hat{i}-14\hat{j}-20\hat{k}}{\sqrt{510}}$
2. $\frac{7}{2}\hat{i}-\hat{j}-2\hat{k}$
3. $\frac{14\hat{i}-4\hat{j}-8\hat{k}}{\sqrt{69}}$

Q. If $4\hat{i}+7\hat{j}+8\hat{k},2\hat{i}+3\hat{j}+4\hat{k}$ and $2\hat{i}+5\hat{j}+7\hat{k}$ are the position vectors of the vertices A, B and C respectively of triangle ABC. The position vector of the point where the bisector of angle A meets BC is

1. $\frac{2}{3}(6\hat{i}+8\hat{j}+6\hat{k})$
2. $\frac{1}{3}(6\hat{i}+13\hat{j}+18\hat{k})$
3. $\frac{2}{3}(-6\hat{i}-8\hat{j}-6\hat{k})$
4. $\frac{1}{3}(5\hat{i}+12\hat{k})$

Q. One vertex of the equilateral triangle with centroid at the origin and one side as $x+y-2=0$ is

1. $(-1,-1)$
2. $(2,2)$
3. $(-2,-2)$
4. Non4 of these