Question

Find the vector equation of a line passing through two points (1,0,4) and (6,-3,1).

(a) \((1+5λ) \hat{i}-λ\hat{j}+(4-3λ) \hat{k}\)

(b) \((1+5λ) \hat{i}-3λ\hat{j}+(7-3λ) \hat{k}\)

(c) \((1+λ) \hat{i}+λ\hat{j}+(8-3λ) \hat{k}\)

(d) \((1+5λ) \hat{i}-3λ\hat{j}+(4-3λ) \hat{k}\)

I have been asked this question in an interview.

Question is taken from Three Dimensional Geometry in portion Three Dimensional Geometry of Mathematics – Class 12

Answer 1

The correct choice is (d) \((1+5λ) \hat{i}-3λ\hat{j}+(4-3λ) \hat{k}\)

Best explanation: Consider the points A(1,0,4) and B(6,-3,1)

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

∴\(\vec{a}=\hat{i}+4\hat{k}\)

\(\vec{b}=6\hat{i}-3\hat{j}+\hat{k}\)

∴\(\vec{r}=\hat{i}+4\hat{k}+λ(6\hat{i}-3\hat{j}+\hat{k}-(\hat{i}+4\hat{k}))\)

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

=\((1+5λ) \hat{i}-3λ\hat{j}+(4-3λ) \hat{k}\)