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Question
The equation of the line passing through the points \[a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} \text{ and } b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k} \] is
Options
\[\overrightarrow{r} = \left( a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} \right) + \lambda \left( b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k} \right)\]
\[\overrightarrow{r} = \left( a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} \right) - t \left( b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k} \right)\]
\[\overrightarrow{r} = a_1 \left( 1 - t \right) \hat{i} + a_2 \left( 1 - t \right) \hat{j} + a_3 \left( 1 - t \right) \hat{k} + t \left( b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k} \right)\]
none of these
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Solution
\[\overrightarrow{r} = a_1 \left( 1 - t \right) \hat{i} + a_2 \left( 1 - t \right) \hat{j} + a_3 \left( 1 - t \right) \hat{k} + t \left( b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k} \right)\]
Equation of the line passing through the points having position vectors
\[a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} \text{ and } b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k} \] is
\[\overrightarrow{r} = \left( a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} \right) + t\left\{ \left( b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k} \right) - \left( a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} \right) \right\}, \text{ where t is a parameter } \]
\[ = \left( a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} \right) - t\left( a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} \right) + t\left( b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k} \right)\]
\[ = a_1 \left( 1 - t \right) \hat{i} + a_2 \left( 1 - t \right) \hat{j} + a_3 \left( 1 - t \right) \hat{k} + t\left( b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k} \right)\]
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