Question

Find the vector equation of a line passing through two points (-5,3,1) and (4,-3,2).

(a) \((-5+λ) \hat{i}+(3+λ)\hat{j}+(1-λ) \hat{k}\)

(b) \((-5+λ) \hat{i}+(3+6λ)\hat{j}+(1+λ) \hat{k}\)

(c) \((5+7λ) \hat{i}+(8+6λ)\hat{j}+(3-5λ) \hat{k}\)

(d) \((-5+9λ) \hat{i}+(3-6λ)\hat{j}+(1+λ) \hat{k}\)

I have been asked this question at a job interview.

I would like to ask this question from Three Dimensional Geometry topic in section Three Dimensional Geometry of Mathematics – Class 12

Answer 1

Right answer is (d) \((-5+9λ) \hat{i}+(3-6λ)\hat{j}+(1+λ) \hat{k}\)

Explanation: Consider the points A(-5,3,1) and B(4,-3,2)

Let \(\vec{a} \,and\, \vec{b}\) be the position vectors of the points A and B.

∴\(\vec{a}=-5\hat{i}+3\hat{j}+\hat{k}\)

\(\vec{b}=4\hat{i}-3\hat{j}+2\hat{k}\)

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

=-\(5\hat{i}+3\hat{j}+\hat{k}+λ(9\hat{i}-6\hat{j}+\hat{k})\)

=\((-5+9λ) \hat{i}+(3-6λ)\hat{j}+(1+λ) \hat{k}\)