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Question

The points with position vectors $$10i+3j,12j$$ and $$ai+11j$$ are collinear then a equals.


Solution

$$\begin{array}{l} \overrightarrow { A } \left( { 10\widehat { i } +3\widehat { j }  } \right) \, \, \, \, \, \overrightarrow { B } \left( { 12 } \right) \widehat { j } \, \, \, \, \, \overrightarrow { C } \left( { a\widehat { i } +11\widehat { j }  } \right)  \\ \overrightarrow { AB } =-10\widehat { i } +9\widehat { j }  \\ \overrightarrow { AC } =\left( { a-10 } \right) \widehat { i } +8\widehat { j }  \\ \overrightarrow { AB } \, \, and\, \, \, \overrightarrow { AC } \, \, are\, \, collinear. \\ \Rightarrow \frac { { -10 } }{ { a-10 } } =\frac { 9 }{ 8 }  \\ \frac { { 10 } }{ { 10-a } } =\frac { 9 }{ 8 }  \\ 80=90-9a \\ 9a=10 \\ \therefore a=\frac { { 10 } }{ 9 }  \end{array}$$

Mathematics

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