Question

The shortest distance between the lines \( \bar{r} = (3\bar{i} - 5\bar{j} + 2\bar{k}) + t(4\bar{i} + 3\bar{j} - \bar{k}) \) and \( \bar{r} = (\bar{i} + 2\bar{j} - 4\bar{k}) + s(6\bar{i} + 3\bar{j} - 2\bar{k}) \) is

• A. 7
• B. 8
• C. 9
• D. 12

Hint:

Always double-check signs when subtracting vectors and calculating determinants, as these are the most common sources of error in shortest distance problems.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The problem asks for the shortest distance between two skew lines given in vector form. The shortest distance is measured along the line perpendicular to both direction vectors.

Step 2: Key Formula or Approach:
For lines r = a1 + t b1 and r = a2 + s b2, the shortest distance d is:
d = | ( (a2 - a1) · (b1 × b2) ) / |b1 × b2| |

Step 3: Detailed Explanation:
Identify the components:
a1 = 3i - 5j + 2k, b1 = 4i + 3j - k
a2 = i + 2j - 4k, b2 = 6i + 3j - 2k

Calculate a2 - a1:
a2 - a1 = (1-3)i + (2-(-5))j + (-4-2)k = -2i + 7j - 6k

Calculate the cross product b1 × b2:

b₁ × b₂ = | i   j   k |
          | 4   3  -1 |
          | 6   3  -2 |

= i(-6 - (-3)) - j(-8 - (-6)) + k(12 - 18)
= i(-3) - j(-2) + k(-6)
= -3i + 2j - 6k

Calculate the magnitude |b1 × b2|:
√((-3)2 + 22 + (-6)2) = √(9 + 4 + 36) = √49 = 7

Calculate the dot product:
(a2 - a1) · (b1 × b2) = (-2)(-3) + (7)(2) + (-6)(-6) = 56

Calculate the distance:
d = |56| / 7 = 8

Step 4: Final Answer:
The shortest distance is 8.