Question

If the points with position vectors \[10 \hat{i} + 3 \hat{j} , 12 \hat{i} - 5 \hat{j}\text{ and a }\hat{i} + 11 \hat{j}\] are collinear, find the value of a.

Answer 1

Let A, B, C be the points with position vectors \[10 \hat{i} + 3 \hat{j} , 12 \hat{i} - 5 \hat{j} , a \hat{i} + 11 \hat{j}\].
Then, \[\overrightarrow{AB} = \] Position vector of B -  Position vector of A
\[= 12 \hat{i} - 5 \hat{j} - 10 \hat{i} - 3 \hat{j} \]
\[ = 2 \hat{i} - 8 \hat{j}\]
\[\overrightarrow{BC} =\] Position vector of C -  Position vector of B
\[= a \hat{i} + 11 \hat{j} - 12 \hat{i} + 5 \hat{j} \]
\[ = \left( a - 12 \right) \hat{i} + 16 \hat{j} \]
Since,
A, B and C  are collinear.
\[\overrightarrow{AB} = \lambda \overrightarrow{BC} .\]
\[\Rightarrow 2 \hat{i} - 8 \hat{j} = \lambda \left( a - 12 \right) \hat{i} + \lambda16 \hat{j} \]
\[ \Rightarrow 2 = \lambda \left( a - 12 \right), - 8 = \lambda16\]
\[ \Rightarrow 2 = \lambda\left( a - 12 \right), \lambda = - \frac{1}{2}\]
\[ \Rightarrow 2 = - \frac{1}{2}\left( a - 12 \right)\]
\[ \Rightarrow - a + 12 = 4\]
\[ \Rightarrow a = 8\]