Match List-I with List-II.

Where $\lambda $ is an arbitrary constant.

Choose the correct answer from the options given below :

(A)-(I),(B)-(IV),(C)-(II),(D)-(III)

The correct answer is Option (2) → (A)-(I),(B)-(IV),(C)-(II),(D)-(III)

(A) equation → $\vec r=\vec a+λ(\vec b-\vec a)$

$\vec a=-\hat i+2\hat k,\vec b=3\hat i+4\hat j+6\hat k$

$⇒\vec{r}=-\hat{i}+2\hat{k})+\lambda (4\hat{i}+4\hat{j}+4\hat{k})$ (I)

(B) Point $(\hat i-\hat j+4\hat k)$

$\vec v$ || line $(2\hat i+3\hat j-5\hat k)$

$⇒\vec{r}=(\hat{i}-\hat{j}+4\hat{k})+\lambda (2\hat{i}+3\hat{j}-5\hat{k})$ (IV)

(C) point $(\hat i-\hat j+4\hat k)$

line || $\vec v$ || direction ratios of given line

$\vec v=2\hat i+\hat j+3\hat k$

$⇒\vec{r}=(\hat{i}+3\hat{j}+2\hat{k})+\lambda (2\hat{i}+\hat{j}+3\hat{k})$ (II)

(D) point $(-\hat i+2\hat j+3\hat k)$

$\vec v$ || line: $\vec v=2\hat i+\hat j+4\hat k$

$\vec{r}=(-\hat{i}+2\hat{j}+3\hat{k})+\lambda (2\hat{i}+\hat{j}+4\hat{k})$ (III)