Find the vector equation of the line which is passing through the point (2,-3,5) and parallel to the vector \(3\hat{i}+4\hat{j}-2\hat{k}\).

(a) \((2+3λ) \hat{i}+(4λ+3) \hat{j}+(5-λ)\hat{k}\)

(b) \((9+3λ) \hat{i}+(λ-3) \hat{j}+(5-2λ)\hat{k}\)

(c) \((2+3λ) \hat{i}+(4λ-3) \hat{j}+(5-2λ)\hat{k}\)

(d) \((7+λ) \hat{i}+(4λ+3) \hat{j}+(5-2λ)\hat{k}\)

This question was posed to me in an online interview.

This key question is from Three Dimensional Geometry topic in division Three Dimensional Geometry of Mathematics – Class 12

NCERT Solutions for Subject Clas 12 Math Select the correct answer from above options

Interview Questions and Answers, Database Interview Questions and Answers for Freshers and Experience

The correct answer is (c) (2+3λ) \hat{i}+(4λ-3) \hat{j}+(5-2λ)\hat{k}

To explain: Given that the line is passing through the point (2,-3,5). Therefore, the position vector of the line is \vec{a}=2\hat{i}-3\hat{j}+5\hat{k}.

Also given that, the line is parallel to a vector \vec{b}=3\hat{i}+4\hat{j}-2\hat{k}.

We know that, the equation of line passing through a point and parallel to vector is given by \vec{r}=\vec{a}+λ\vec{b}, where λ is a constant.

∴\vec{r}=2\hat{i}-3\hat{j}+5\hat{k}+λ(3\hat{i}+4\hat{j}-2\hat{k})

=(2+3λ) \hat{i}+(4λ-3) \hat{j}+(5-2λ)\hat{k}