Show that the point A (1, – 1, 3),
Question:

Show that the point A (1, – 1, 3), B (2, – 4, 5) and (5, – 13, 11) are collinear.

Solution:

Given points are $A(1,-1,3), B(2,-4,5)$ and $(5,-13,11)$.

To prove collinear,

$\mathrm{AB}=\sqrt{(1-2)^{2}+(-1+4)^{2}+(3-5)^{2}}=\sqrt{1+9+4}=\sqrt{14}$

$\mathrm{BC}=\sqrt{(2-5)^{2}+(-4+13)^{2}+(5-11)^{2}}=\sqrt{9+81+36}=3 \sqrt{14}$

$\mathrm{AC}=\sqrt{(1-5)^{2}+(-1+13)^{2}+(3-11)^{2}}=\sqrt{16+144+64}=4 \sqrt{14}$

$\therefore \mathrm{AB}+\mathrm{BC}=\sqrt{14}+3 \sqrt{14}$

$=4 \sqrt{14}$

$=\mathrm{AC}$

$\therefore$ Points A, B and C are collinear.

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