Question

The equation of the line passing through the points $a_1\hat{i}+a_2\hat{j}+a_3\hat{k}\mathrm{and}{b}_1\hat{i}+b_2\hat{j}+b_3\hat{k}$ is
(a) $\overrightarrow{r}=\left(a_1\hat{i}+a_2\hat{j}+a_3\hat{k}\right)+\lambda \left(b_1\hat{i}+b_2\hat{j}+b_3\hat{k}\right)$

(b) $\overrightarrow{r}=\left(a_1\hat{i}+a_2\hat{j}+a_3\hat{k}\right)-t\left(b_1\hat{i}+b_2\hat{j}+b_3\hat{k}\right)$

(c) $\overrightarrow{r}=a_1\left(1-t\right)\hat{i}+a_2\left(1-t\right)\hat{j}+a_3\left(1-t\right)\hat{k}+t\left(b_1\hat{i}+b_2\hat{j}+b_3\hat{k}\right)$

(d) none of these

Answer 1

(c) $\overrightarrow{r}=a_1\left(1-t\right)\hat{i}+a_2\left(1-t\right)\hat{j}+a_3\left(1-t\right)\hat{k}+t\left(b_1\hat{i}+b_2\hat{j}+b_3\hat{k}\right)$

Equation of the line passing through the points having position vectors $a_1\hat{i}+a_2\hat{j}+a_3\hat{k}\mathrm{and}{b}_1\hat{i}+b_2\hat{j}+b_3\hat{k}$ is

$$\begin{gathered} \overrightarrow{r}=\left(a_1\hat{i}+a_2\hat{j}+a_3\hat{k}\right)+t\left\{\left(b_1\hat{i}+b_2\hat{j}+b_3\hat{k}\right)-\left(a_1\hat{i}+a_2\hat{j}+a_3\hat{k}\right)\right\},\mathrm{where}t\mathrm{is}\mathrm{a}\mathrm{parameter} \\ =\left(a_1\hat{i}+a_2\hat{j}+a_3\hat{k}\right)-t\left(a_1\hat{i}+a_2\hat{j}+a_3\hat{k}\right)+t\left(b_1\hat{i}+b_2\hat{j}+b_3\hat{k}\right) \\ =a_1\left(1-t\right)\hat{i}+a_2\left(1-t\right)\hat{j}+a_3\left(1-t\right)\hat{k}+t\left(b_1\hat{i}+b_2\hat{j}+b_3\hat{k}\right) \end{gathered}$$