The equation of the line passing through the points ${a_1}{\bf{\hat i}} + {a_2}{\bf{\hat j}} + {a_3}{\bf{\hat k}}\;{\rm{and}}\;{b_1}{\bf{\hat i}} + {b_2}{\bf{\hat j}} + {b_3}{\bf{\hat k}}$ is

${a_1}\left( {1 - t} \right){\bf{\hat i}} + {a_2}\left( {1 - t} \right){\bf{\hat j}} + {a_3}\left( {1 - t} \right){\bf{\hat k}} + \left( {\;{b_1}{\bf{\hat i}} + {b_2}{\bf{\hat j}} + {b_3}{\bf{\hat k}}} \right)t$

$\left( {{a_1}{\bf{\hat i}} + {a_2}{\bf{\hat j}} + {a_3}{\bf{\hat k}}} \right) + t\left( {\;{b_1}{\bf{\hat i}} + {b_2}{\bf{\hat j}} + {b_3}{\bf{\hat k}}} \right)$

$\left( {\;{a_1}{\bf{\hat i}} + {a_2}{\bf{\hat j}} + {a_3}{\bf{\hat k}}\;} \right) - t\left( {{b_1}{\bf{\hat i}} + {b_2}{\bf{\hat j}} + {b_3}{\bf{\hat k}}} \right)$

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JEE MATHEMATICS MCQ: ADDITION OF VECTORS

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The equation of the line passing through the points

{a_1}\mathbf{\hat i} + {a_2}\mathbf{\hat j} + {a_3}\mathbf{\hat k}\;\mathrm{and}\;{b_1}\mathbf{\hat i} + {b_2}\mathbf{\hat j} + {b_3}\mathbf{\hat k}

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Correct option: C

{a_1}\left( {1 - t} \right)\mathbf{\hat i} + {a_2}\left( {1 - t} \right)\mathbf{\hat j} + {a_3}\left( {1 - t} \right)\mathbf{\hat k} + \left( {\;{b_1}\mathbf{\hat i} + {b_2}\mathbf{\hat j} + {b_3}\mathbf{\hat k}} \right)t

Option C is correct because it represents the parametric equation of a line passing through two points in vector form.

JEE MATHEMATICS MCQ: ADDITION OF VECTORS

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