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  4. The points (1,2,3), (- 1, - 1, - 1) and (3, 5, 7) are the vertices of

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Q. The points $(1,2,3)$, $(- 1, - 1, - 1)$ and $(3, 5, 7)$ are the vertices of

Introduction to Three Dimensional Geometry

Solution:

Here, the points are $A(1, 2, 3)$, $B(- 1, - 1, - 1)$ and $C(3, 5, 7)$.
$\left|AB\right| = \sqrt{2^{2}+3^{2}+4^{2}} = \sqrt{29}$,
$\left|BC\right| = \sqrt{4^{2}+6^{2}+8^{2}}=\sqrt{116} = 2\sqrt{29}$
and $\left|CA\right| = \sqrt{2^{2}+3^{2}+4^{2}}=\sqrt{29}$.
So, $|BC| = |AB| + |CA|$
$\Rightarrow $ The points $A$, $B$, $C$ are collinear.