Using vector method, prove that the point is collinear: A(1, 2, 7… - Vidyadip

Question

Using vector method, prove that the point is collinear:
A(1, 2, 7), B(2, 6, 3) and C(3, 10, -1)

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Answer

Given the points A(1, 2, 7), B(2, 6, 3) and C(3, 10, -1). Then,

$\overrightarrow{\text{AB}}=$ Position vector of B - Position vector of A

$=2\hat{\text{i}}+6\hat{\text{j}}+3\hat{\text{k}}-\hat{\text{i}}-2\hat{\text{j}}-7\hat{\text{k}}$

$=\hat{\text{i}}+4\hat{\text{j}}-4\hat{\text{k}}$

$\overrightarrow{\text{BC}}=$ Position vector of C - Position vector of B

$=3\hat{\text{i}}+10\hat{\text{j}}-\hat{\text{k}}-2\hat{\text{i}}-6\hat{\text{j}}-3\hat{\text{k}}$

$=\hat{\text{i}}+4\hat{\text{j}}-4\hat{\text{k}}$

$\therefore\ \overrightarrow{\text{AB}}=\overrightarrow{\text{BC}}$

So, $\overrightarrow{\text{AB}}$ and $\overrightarrow{\text{BC}}$ are parallel vectors. But B is a point common to them.

Hence, the given points A, B, and C are collinear.

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