Question

Find the vector equation of the line which is passing through the point (1, -4, 4) and parallel to the vector \(2\hat{i}-5\hat{j}+2\hat{k}\).

(a) \((1+2λ) \hat{i}-(4+5λ) \hat{j}+(4+2λ) \hat{j}\)

(b) \((1+2λ) \hat{i}-(4+5λ) \hat{j}+2λ \hat{j}\)

(c) \((1-2λ) \hat{i}+(4+5λ) \hat{j}+(4+2λ) \hat{j}\)

(d) \((8+λ) \hat{i}-(4-5λ) \hat{j}+(7-λ) \hat{j}\)

I have been asked this question in semester exam.

The doubt is from Three Dimensional Geometry in portion Three Dimensional Geometry of Mathematics – Class 12

Answer 1

The correct option is (a) \((1+2λ) \hat{i}-(4+5λ) \hat{j}+(4+2λ) \hat{j}\)

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

The position of vector of the point (1,-4,4) is given by \(\vec{a}=\hat{i}-4\hat{j}+4\hat{k}\)

And \(\vec{b}=2\hat{i}-5\hat{j}+2\hat{k}\)

∴\(\vec{r}=\vec{a}+λ\vec{b}\)

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

=\((1+2λ) \hat{i}-(4+5λ) \hat{j}+(4+2λ) \hat{j}\)