Consider a three dimensional Cartesian system with origin at O and three rectangular coordinates axe

Question

Consider a three dimensional Cartesian system with origin at O and three rectangular coordinates axes $\mathrm{x}, \mathrm{y}$ and z -axis. Suppose that the distance between two points P and Q in the space having their coordinates $\left(x_1, y_1, z_1\right)$ and $\left(x_2, y_2, z_2\right)$ respectively be defined by the following formula $d(P, Q)=\left|x_2-x_1\right|+\left|y_2-y_1\right|+\left|z_2-z_1\right|$.
Although the formula of distance between two points has been defined in a new way, yet the other definition remain same (like section formula, direction cosines etc.). So, in general equations of straight line in space remain unchanged. Now answer the following questions.

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See detailed solution for explanation.