If a line is passing through two points \(A(x_1,y_1,z_1)\) and \(B(x_2,y_2,z_2)\) then which of the following is the vector equation of the line?

(a) \(\vec{r}=\vec{a}+λ(\vec{b}+\vec{a})\)

(b) \(\vec{r}=\vec{a}+λ(\vec{a}-\vec{b})\)

(c) \(\vec{r}=λ\vec{a}+(\vec{b}-\vec{a})\)

(d) \(\vec{r}=\vec{a}+λ(\vec{b}-\vec{a})\)

The question was posed to me in my homework.

My enquiry is from Three Dimensional Geometry in section Three Dimensional Geometry of Mathematics – Class 12

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Right choice is (d) \vec{r}=\vec{a}+λ(\vec{b}-\vec{a})

To explain: Let \vec{a} \,and\, \vec{b} are the position vectors of two points A(x_1,y_1,z_1) and B(x_2,y_2,z_2)

Then the vector equation of the line is given by the formula passing through two points will be given by

\vec{r}=\vec{a}+λ(\vec{b}-\vec{a}), λ∈R