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Question
Using distance formula prove that the following points are collinear:
A(4, –3, –1), B(5, –7, 6) and C(3, 1, –8)
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Solution
AB =\[\sqrt{\left( 5 - 4 \right)^2 + \left( - 7 + 3 \right)^2 + \left( 6 + 1 \right)^2}\]
\[= \sqrt{\left( 1 \right)^2 + \left( - 4 \right)^2 + \left( 7 \right)^2}\]
\[ = \sqrt{1 + 16 + 49}\]
\[ = \sqrt{66}\]
BC =\[\sqrt{\left( 3 - 5 \right)^2 + \left( 1 + 7 \right)^2 + \left( - 8 - 6 \right)^2}\]
=\[\sqrt{\left( - 2 \right)^2 + \left( 8 \right)^2 + \left( - 14 \right)^2}\]
\[= \sqrt{4 + 64 + 196}\]
\[ = \sqrt{264}\]
\[ = 2\sqrt{66}\]
AC =\[\sqrt{\left( 3 - 4 \right)^2 + \left( 1 + 3 \right)^2 + \left( - 8 + 1 \right)^2}\]
\[= \sqrt{\left( - 1 \right)^2 + \left( 4 \right)^2 + \left( - 7 \right)^2}\]
\[ = \sqrt{1 + 16 + 49}\]
\[ = \sqrt{66}\]
\[\text{ Here } , AB + AC = \sqrt{66} + \sqrt{66}\]
\[ = 2\sqrt{66}\]
\[ = BC\]
Hence, the points are collinear.
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