The distance between the points of the line \(\vec{r} = (\hat{\imath}-2\hat{\jmath}+3\hat{k})+\lambda(2\hat{\imath}-\hat{\jmath}+2\hat{k})\) given by \(\lambda = -1\) and \(\lambda = 3\) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 12
Putting the two values of \(\lambda\) in the equation:
\(\lambda = -1:\quad (1-2,\ -2+1,\ 3-2) = (-1, -1, 1)\)
\(\lambda = 3:\quad (1+6,\ -2-3,\ 3+6) = (7, -5, 9)\)
The distance between them is
\(\sqrt{8^2+(-4)^2+8^2} = \sqrt{144} = 12\)
Shortcut: The two points differ by \((3-(-1))\vec{b} = 4\vec{b}\), so the distance is \(4|\vec{b}| = 4\sqrt{4+1+4} = 4\times 3 = 12\).