Match the List-I with List-II
| List-I | List-II | ||
| Line | Direction cosines | ||
| (A) | \( \vec{r}=\left(2\hat{\imath}-\hat{\jmath}+\hat{k}\right)+\lambda\left(3\hat{\imath}+2\hat{k}\right) \) | (I) | \( \left\langle\displaystyle\frac{2}{\sqrt{5}},\displaystyle\frac{1}{\sqrt{5}},0\right\rangle \) |
| (B) | \( \displaystyle\frac{x-1}{2}=\displaystyle\frac{3-y}{1}=\displaystyle\frac{2z-1}{4} \) | (II) | \( \left\langle-\displaystyle\frac{1}{3},\displaystyle\frac{2}{3},-\displaystyle\frac{2}{3}\right\rangle \) |
| (C) | \( \displaystyle\frac{x+1}{1}=\displaystyle\frac{2y+1}{1},\ z=-1 \) | (III) | \( \left\langle\displaystyle\frac{3}{\sqrt{13}},0,\displaystyle\frac{2}{\sqrt{13}}\right\rangle \) |
| (D) | \( \displaystyle\frac{1-x}{1}=\displaystyle\frac{y}{2}=\displaystyle\frac{1-z}{2} \) | (IV) | \( \left\langle\displaystyle\frac{2}{3},-\displaystyle\frac{1}{3},\displaystyle\frac{2}{3}\right\rangle \) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) - (III), (B) - (IV), (C) - (I), (D) - (II)
In the standard form \( \frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c} \), with coefficient +1 on x, y and z, the numbers a, b, c are direction ratios. Dividing them by \( \sqrt{a^{2}+b^{2}+c^{2}} \) gives the direction cosines.
(A) \( \vec{r}=\left(2\hat{\imath}-\hat{\jmath}+\hat{k}\right)+\lambda\left(3\hat{\imath}+2\hat{k}\right) \)
This matches (III) \( \left\langle\frac{3}{\sqrt{13}},0,\frac{2}{\sqrt{13}}\right\rangle \).
\( \displaystyle \text{Direction ratios }3,\ 0,\ 2 \)
\( \displaystyle \sqrt{9+0+4}=\sqrt{13} \)
(B) \( \frac{x-1}{2}=\frac{3-y}{1}=\frac{2z-1}{4} \)
This matches (IV) \( \left\langle\frac{2}{3},-\frac{1}{3},\frac{2}{3}\right\rangle \).
\( \displaystyle \frac{x-1}{2}=\frac{y-3}{-1}=\frac{z-\frac{1}{2}}{2} \)
\( \displaystyle \text{Direction ratios }2,\ -1,\ 2 \)
\( \displaystyle \sqrt{4+1+4}=3 \)
(C) \( \frac{x+1}{1}=\frac{2y+1}{1},\ z=-1 \)
This matches (I) \( \left\langle\frac{2}{\sqrt{5}},\frac{1}{\sqrt{5}},0\right\rangle \).
\( \displaystyle \frac{x+1}{1}=\frac{y+\frac{1}{2}}{\frac{1}{2}},\quad z=-1\ \text{(constant, so the third ratio is 0)} \)
\( \displaystyle \text{Direction ratios }1,\ 1/2,\ 0,\ \text{that is }2,\ 1,\ 0 \)
\( \displaystyle \sqrt{4+1+0}=\sqrt{5} \)
(D) \( \frac{1-x}{1}=\frac{y}{2}=\frac{1-z}{2} \)
This matches (II) \( \left\langle-\frac{1}{3},\frac{2}{3},-\frac{2}{3}\right\rangle \).
\( \displaystyle \frac{x-1}{-1}=\frac{y-0}{2}=\frac{z-1}{-2} \)
\( \displaystyle \text{Direction ratios }-1,\ 2,\ -2 \)
\( \displaystyle \sqrt{1+4+4}=3 \)
So the matching is (A)-(III), (B)-(IV), (C)-(I), (D)-(II).
Tip: Before reading direction ratios, make the coefficients of x, y and z equal to +1: \( 3-y \) gives the ratio −1, and \( 2z-1 \) must be divided by 2 first.