Match List - I with List - II.
| List - I | List - II | ||
| (A) | Cartesian equation of y axis | (I) | \(\displaystyle\frac{x-0}{0}=\frac{y-0}{0}=\frac{z-0}{1}\) |
| (B) | Vector equation of line \(\displaystyle\frac{x-5}{3}=\frac{y+4}{7}=\frac{z-6}{2}\) | (II) | \(\vec{r}=\left(5\hat{\imath}-4\hat{\jmath}+6\hat{k}\right)+\lambda\left(3\hat{\imath}+7\hat{\jmath}+2\hat{k}\right)\) |
| (C) | Cartesian equation of z axis | (III) | \(\displaystyle\frac{x-0}{0}=\frac{y-0}{1}=\frac{z-0}{0}\) |
| (D) | Vector equation of line \(\displaystyle\frac{x-3}{5}=\frac{y-7}{-4}=\frac{z-2}{6}\) | (IV) | \(\vec{r}=\left(3\hat{\imath}+7\hat{\jmath}+2\hat{k}\right)+\mu\left(5\hat{\imath}-4\hat{\jmath}+6\hat{k}\right)\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) - (III), (B) - (II), (C) - (I), (D) - (IV)
A line through the point \((x_1, y_1, z_1)\) with direction ratios a, b, c has the Cartesian form \(\frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c}\) and the vector form \(\vec{r} = (x_1\hat{\imath}+y_1\hat{\jmath}+z_1\hat{k})+\lambda(a\hat{\imath}+b\hat{\jmath}+c\hat{k})\).
(A) The y axis passes through the origin with direction ratios 0, 1, 0:
\( \displaystyle \frac{x-0}{0}=\frac{y-0}{1}=\frac{z-0}{0} \quad \text{(III)} \)
(B) Point (5, −4, 6), direction ratios 3, 7, 2:
\(\vec{r}=\left(5\hat{\imath}-4\hat{\jmath}+6\hat{k}\right)+\lambda\left(3\hat{\imath}+7\hat{\jmath}+2\hat{k}\right) \quad \text{(II)}\)
(C) The z axis passes through the origin with direction ratios 0, 0, 1:
\( \displaystyle \frac{x-0}{0}=\frac{y-0}{0}=\frac{z-0}{1} \quad \text{(I)} \)
(D) Point (3, 7, 2), direction ratios 5, −4, 6:
\(\vec{r}=\left(3\hat{\imath}+7\hat{\jmath}+2\hat{k}\right)+\mu\left(5\hat{\imath}-4\hat{\jmath}+6\hat{k}\right) \quad \text{(IV)}\)
So the match is (A)-(III), (B)-(II), (C)-(I), (D)-(IV).