Match List-I with List-II
| List-I | List-II | ||
| (A) | The value of \(\lambda\) for which \(2\hat{\imath}+\lambda\hat{\jmath}-4\hat{k}\) and \(-\hat{\imath}+3\hat{\jmath}+2\hat{k}\) are collinear | (I) | \(\displaystyle\frac{2}{3}\) |
| (B) | The value of t for which the points with position vectors \(\hat{\imath}+2\hat{\jmath}\), \(2\hat{\imath}+3\hat{\jmath}+\hat{k}\) and \(4\hat{\imath}+t\hat{\jmath}+3\hat{k}\) are collinear | (II) | −3 |
| (C) | The positive value of c for which \(\displaystyle\frac{1}{3}\hat{\imath}+\frac{2}{3}\hat{\jmath}+c\hat{k}\) is a unit vector | (III) | −6 |
| (D) | The value of k for which \(k(2\hat{\imath}-\hat{\jmath}+2\hat{k})\) has magnitude 9 and is opposite in direction to \(2\hat{\imath}-\hat{\jmath}+2\hat{k}\) | (IV) | 5 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A)-(III), (B)-(IV), (C)-(I), (D)-(II)
(A) The value of \(\lambda\) for which \(2\hat{\imath}+\lambda\hat{\jmath}-4\hat{k}\) and \(-\hat{\imath}+3\hat{\jmath}+2\hat{k}\) are collinear
This matches (III) −6.
Collinear vectors have proportional components:
\( \displaystyle \frac{2}{-1} = \frac{\lambda}{3} = \frac{-4}{2} = -2 \)
\(\lambda = 3\times(-2) = -6\)
(B) The value of t for which the points with position vectors \(\hat{\imath}+2\hat{\jmath}\), \(2\hat{\imath}+3\hat{\jmath}+\hat{k}\) and \(4\hat{\imath}+t\hat{\jmath}+3\hat{k}\) are collinear
This matches (IV) 5.
Call the points P, Q, R. Then
\(\overrightarrow{PQ} = \hat{\imath}+\hat{\jmath}+\hat{k}\)
\(\overrightarrow{PR} = 3\hat{\imath}+(t-2)\hat{\jmath}+3\hat{k}\)
For collinear points \(\overrightarrow{PR} = 3\,\overrightarrow{PQ}\), as the \(\hat{\imath}\)- and \(\hat{k}\)-components show:
\(t-2 = 3\), so \(t = 5\)
(C) The positive value of c for which \(\displaystyle\frac{1}{3}\hat{\imath}+\frac{2}{3}\hat{\jmath}+c\hat{k}\) is a unit vector
This matches (I) \(\frac{2}{3}\).
A unit vector has magnitude 1:
\( \displaystyle \frac{1}{9}+\frac{4}{9}+c^2 = 1 \)
\( \displaystyle c^2 = \frac{4}{9} \), so \( c = \frac{2}{3} \)
(D) The value of k for which \(k(2\hat{\imath}-\hat{\jmath}+2\hat{k})\) has magnitude 9 and is opposite in direction to \(2\hat{\imath}-\hat{\jmath}+2\hat{k}\)
This matches (II) −3.
\(|2\hat{\imath}-\hat{\jmath}+2\hat{k}| = \sqrt{4+1+4} = 3\)
\(|k|\times 3 = 9\), so \(|k| = 3\)
The opposite direction needs a negative scalar, so \(k = -3\).