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

Question

Using vector method, prove that the point is collinear:
A(2, -1, 3), B(4, 3, 1) and C(3, 1, 2)

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Answer

Given the points A(2, -1, 3), B(4, 3, 1) and C(3, 1, 2). Then, $\overrightarrow{\text{AB}}=$ Position vector of B - Position vector of A$=4\hat{\text{i}}+3\hat{\text{j}}+\hat{\text{k}}-2\hat{\text{i}}+\hat{\text{j}}-3\hat{\text{k}}$
$=2\hat{\text{i}}+4\hat{\text{j}}-2\hat{\text{k}}$
$=-2\big(-\hat{\text{i}}-2\hat{\text{j}}+\hat{\text{k}}\big)$$\overrightarrow{\text{BC}}=$ Position vector of C - Position vector of B
$=3\hat{\text{i}}+\hat{\text{j}}+2\hat{\text{k}}-4\hat{\text{i}}-3\hat{\text{j}}-\hat{\text{k}}$
$=-\hat{\text{i}}-2\hat{\text{j}}+\hat{\text{k}}$
$\therefore\ \overrightarrow{\text{AB}}=-2\overrightarrow{\text{BC}}$
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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