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.

Answer 1

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.