Question

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

Answer 1

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.

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

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