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(B) Given position vectors are $\vec{a} = \hat{i} + 2\hat{j} - 5\hat{k}$,$\vec{b} = 3\hat{i} + 6\hat{j} + 3\hat{k}$,$\vec{c} = \frac{17}{5}\hat{i} + \

Vedclass · Accepted Answer

$\frac{\vec{b}+2 \vec{d}}{3}$ is the position vector of a point which divides $PR$ internally in the ratio $5: 4$

Vedclass · Answer

(B) Given position vectors are $\vec{a} = \hat{i} + 2\hat{j} - 5\hat{k}$,$\vec{b} = 3\hat{i} + 6\hat{j} + 3\hat{k}$,$\vec{c} = \frac{17}{5}\hat{i} + \frac{16}{5}\hat{j} + 7\hat{k}$,and $\vec{d} = 2\hat{i} + \hat{j} + \hat{k}$.Consider the expression $\frac{\vec{b} + 2\vec{d}}{3} = \frac{(3\hat{i} + 6\hat{j} + 3\hat{k}) + 2(2\hat{i} + \hat{j} + \hat{k})}{3} = \frac{7\hat{i} + 8\hat{j} + 5\hat{k}}{3}$.Now,consider the point dividing $PR$ internally in the ratio $5:4$. Its position vector is $\frac{5\vec{c} + 4\vec{a}}{5+4} = \frac{5(\frac{17}{5}\hat{i} + \frac{16}{5}\hat{j} + 7\hat{k}) + 4(\hat{i} + 2\hat{j} - 5\hat{k})}{9} = \frac{(17\hat{i} + 16\hat{j} + 35\hat{k}) + (4\hat{i} + 8\hat{j} - 20\hat{k})}{9} = \frac{21\hat{i} + 24\hat{j} + 15\hat{k}}{9} = \frac{7\hat{i} + 8\hat{j} + 5\hat{k}}{3}$.Since both expressions yield the same vector,the point represented by $\frac{\vec{b} + 2\vec{d}}{3}$ divides $PR$ internally in the ratio $5:4$.For option $D$,$|\vec{b} 	imes \vec{d}|^2 = |\vec{b}|^2 |\vec{d}|^2 - (\vec{b} \cdot \vec{d})^2 = (3^2 + 6^2 + 3^2)(2^2 + 1^2 + 1^2) - (3(2) + 6(1) + 3(1))^2 = (9 + 36 + 9)(4 + 1 + 1) - (6 + 6 + 3)^2 = 54 	imes 6 - (15)^2 = 324 - 225 = 99$. Thus,option $D$ is incorrect.Therefore,the correct statement is $B$.

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