Show that the points whose position vectors are as given below ar… - Vidyadip

Question

Show that the points whose position vectors are as given below are collinear:
$3\hat{\text{i}}-2\hat{\text{j}}+4\hat{\text{k}},\ \hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}$ and $-\hat{\text{i}}+4\hat{\text{j}}-2\hat{\text{k}}$

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Answer

Let the points be A, B and C with position vectors $3\hat{\text{i}}-2\hat{\text{j}}+4\hat{\text{k}},\ \hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}$ and $-\hat{\text{i}}+4\hat{\text{j}}-2\hat{\text{k}}$ respectively. Then,

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

$=\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}-3\hat{\text{i}}+2\hat{\text{j}}-4\hat{\text{k}}$

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

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

$=-\hat{\text{i}}+4\hat{\text{j}}-2\hat{\text{k}}-\hat{\text{i}}-\hat{\text{j}}-\hat{\text{k}}$

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

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

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

Hence, A, B, and C are collinear.

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