First slide

Straight lines in 3D

Question

The shortest distance between the lines $r = (4 i - j) + λ (i + 2 j - 3 k)$ and $r = (i - j + 2 k) + μ (2 i + 4 j - 5 k)$ is

Moderate

A

6
B

$\sqrt{5}$
C

$6 / \sqrt{5}$
D

$6 \sqrt{5}$

Solution

$\begin{array}{l} d = \frac{[(4 i - j) - (i - j + 2 k)] \cdot (i + 2 j - 3 k) \times (2 i + 4 j - 5 k)}{| (i + 2 j - 3 k) \times (2 i + 4 j - 5 k) |} \\ = \frac{(3 i - 2 k) \cdot (2 i - j)}{| 2 i - j |} = \frac{6}{\sqrt{5}} . \end{array}$

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