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Question
Show that the points whose position vectors are \[- 2 \hat{i} + 3 \hat{j} , \hat{i} + 2 \hat{j} + 3 \hat{k} \text{ and } 7 \text{ i} - \text{ k} \] are collinear.
Solution
Let the given points be P, Q and R and let their position vectors be \[\overrightarrow{a} , \overrightarrow{b} \text { and } \overrightarrow{c} , \text{ respectively } . \]
\[\overrightarrow{a} = - 2 \hat{i} + 3 \hat{j} \]
\[ \overrightarrow{b} = \hat{i} + 2 \hat{j} + 3 \hat{k} \]
\[ \overrightarrow{c} = 7 \hat{i} + 9 \hat{k} \]
Vector equation of line passing through P and Q is
\[\overrightarrow{r} = \overrightarrow{a} + \lambda\left( \overrightarrow{b} - \overrightarrow{a} \right)\]
\[ \Rightarrow \overrightarrow{r} = \left( - 2 \hat{i} + 3 \hat{j}
\right) + \lambda\left\{ \left( \hat{i} + 2 \hat{j} + 3 \hat{k} \right) - \left( - 2 \hat{i} + 3 \hat{j} \right) \right\}\]
\[ \Rightarrow \vec{r} = \left( - 2 \hat{i} + 3 \hat{j} \right) + \lambda\left( 3 \hat{i} - \hat{j} + 3 \hat{k} \right) . . . (1)\]
If points P, Q and R are collinear, then point R must satisfy (1).
\[\text{ Replacing } \overrightarrow{r} \text{ by } \overrightarrow{c} = 7 \hat{i} + 9 \hat{k} \text{ in } (1), \text { we get } \]
\[7 \hat{i} + 9 \hat{k} = \left( - 2 \hat{i} + 3 \hat{j} \right) + \lambda\left( 3 \hat{i} - \hat{j} + 3 \hat{k} \right)\]
Comparing the coefficients of \[\hat{i} , \hat{j} \text{ and } \hat{k}\]
we get
\[7 = - 2 + 3\lambda, 0 = 3 - \lambda, 9 = 3\lambda\]
∴ \[\lambda\] = 3
These three equations are consistent, i.e. they give the same value of \[\lambda\] Hence, the given three points are collinear.
Disclaimer: The question given in the book has a minor error. The third vectors should be \[7 \hat{i }+ 9 \hat{k} \] The solution here is created accordingly.
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