Consider the equation of the line \(\vec{r}= -\hat{\imath} + 2\hat{k} + \mu(4\hat{\imath} -\hat{\jmath} + 2\hat{k})\),
Match List-I with List-II
| List-I | List-II | ||
| (A) | It passes through the point | (I) | 4 , -1, 2 |
| (B) | Its direction ratios are | (II) | \(\displaystyle\frac{4}{\sqrt{21}},\frac{-1}{\sqrt{21}},\frac{2}{\sqrt{21}}\) |
| (C) | Its Cartesian form is | (III) | (-1, 0, 2) |
| (D) | Its direction cosines are | (IV) | \(\displaystyle\frac{x+1}{4}=\frac{y}{-1}=\frac{z-2}{2}\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A) - (III), (B) - (I), (C) - (IV), (D) - (II)
The line is in the form \(\vec{r} = \vec{a} + \mu\vec{b}\), where \(\vec{a}\) is the position vector of a point on the line and \(\vec{b}\) is parallel to the line:
\(\vec{a} = -\hat{\imath} + 0\hat{\jmath} + 2\hat{k},\quad \vec{b} = 4\hat{\imath} - \hat{\jmath} + 2\hat{k}\)
(A) It passes through the point
This matches (III) (−1, 0, 2), the point with position vector \(\vec{a}\).
(B) Its direction ratios are
This matches (I) 4, −1, 2, the components of \(\vec{b}\).
(C) Its Cartesian form is
This matches (IV). Through \((-1, 0, 2)\) with direction ratios 4, −1, 2:
\(\displaystyle \frac{x + 1}{4} = \frac{y - 0}{-1} = \frac{z - 2}{2}\)
(D) Its direction cosines are
This matches (II). Dividing the direction ratios by \(|\vec{b}|\):
\(\displaystyle |\vec{b}| = \sqrt{16 + 1 + 4} = \sqrt{21}\)
\(\displaystyle \frac{4}{\sqrt{21}},\ \frac{-1}{\sqrt{21}},\ \frac{2}{\sqrt{21}}\)
So the matching is (A) - (III), (B) - (I), (C) - (IV), (D) - (II).