If 4 hat{i}+7 hat{j}+8 hat{k},2 hat{i}+3 hat{ | vedclass

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(D) The angle bisector theorem states that the bisector of $\angle A$ divides the opposite side $BC$ in the ratio of the lengths of the adjacent sides

Vedclass · Accepted Answer

$2 \hat{i}+\frac{13}{3} \hat{j}+6 \hat{k}$

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(D) The angle bisector theorem states that the bisector of $\angle A$ divides the opposite side $BC$ in the ratio of the lengths of the adjacent sides $AB$ and $AC$,i.e.,in the ratio $c:b$,where $c = |AB|$ and $b = |AC|$.First,calculate the lengths of sides $AB$ and $AC$:$AB = |(2-4)\hat{i} + (3-7)\hat{j} + (4-8)\hat{k}| = |-2\hat{i} - 4\hat{j} - 4\hat{k}| = \sqrt{(-2)^2 + (-4)^2 + (-4)^2} = \sqrt{4 + 16 + 16} = \sqrt{36} = 6$.$AC = |(2-4)\hat{i} + (5-7)\hat{j} + (7-8)\hat{k}| = |-2\hat{i} - 2\hat{j} - 1\hat{k}| = \sqrt{(-2)^2 + (-2)^2 + (-1)^2} = \sqrt{4 + 4 + 1} = \sqrt{9} = 3$.Thus,the ratio is $6:3 = 2:1$.The position vector of the point $D$ dividing $BC$ in ratio $2:1$ is given by the section formula:$\vec{D} = \frac{2\vec{C} + 1\vec{B}}{2+1} = \frac{2(2\hat{i} + 5\hat{j} + 7\hat{k}) + 1(2\hat{i} + 3\hat{j} + 4\hat{k})}{3}$.$\vec{D} = \frac{(4+2)\hat{i} + (10+3)\hat{j} + (14+4)\hat{k}}{3} = \frac{6\hat{i} + 13\hat{j} + 18\hat{k}}{3} = 2\hat{i} + \frac{13}{3}\hat{j} + 6\hat{k}$.Therefore,the correct option is $D$.

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 respectively the position vectors of the vertices $A, B, C$ of $\triangle ABC$,then the position vector of the point where the bisector of angle $A$ meets $BC$ is

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