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(B) Given points $A(1, 2, -1)$ and $B(-1, 0, 1)$. Point $P$ divides the line segment $AB$ externally in the ratio $m_1: m_2 = 1: 2$.For external divis

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$3$

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(B) Given points $A(1, 2, -1)$ and $B(-1, 0, 1)$. Point $P$ divides the line segment $AB$ externally in the ratio $m_1: m_2 = 1: 2$.For external division,the coordinates of $P$ are given by $\left( \frac{m_1 x_2 - m_2 x_1}{m_1 - m_2}, \frac{m_1 y_2 - m_2 y_1}{m_1 - m_2}, \frac{m_1 z_2 - m_2 z_1}{m_1 - m_2} \right)$.Substituting the values:$P = \left( \frac{1(-1) - 2(1)}{1 - 2}, \frac{1(0) - 2(2)}{1 - 2}, \frac{1(1) - 2(-1)}{1 - 2} \right)$$P = \left( \frac{-1 - 2}{-1}, \frac{-4}{-1}, \frac{1 + 2}{-1} \right)$$P = \left( \frac{-3}{-1}, \frac{-4}{-1}, \frac{3}{-1} \right) = (3, 4, -3)$.Now,we find the distance $PQ$ where $Q = (1, 3, -1)$:$PQ = \sqrt{(3 - 1)^2 + (4 - 3)^2 + (-3 - (-1))^2}$$PQ = \sqrt{(2)^2 + (1)^2 + (-2)^2}$$PQ = \sqrt{4 + 1 + 4} = \sqrt{9} = 3$ units.

If $P$ divides the line segment joining the points $A(1, 2, -1)$ and $B(-1, 0, 1)$ externally in the ratio $1: 2$ and $Q = (1, 3, -1)$,then $PQ =$

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