Question

One vertex of a rectangular parallelepiped is at the origin O and the lengths of its edges along x, y and z axes are 3, 4 and 5 units respectively. Let P be the vertex (3, 4, 5). Then the shortest distance between the diagonal OP and an edge parallel to z axis, not passing through O or P is :

• A. \(\frac{16}{5}\)
• B. \(\frac{15}{\sqrt{34}}\)
• C. \(\frac{12}{5}\)
• D. \(\frac{9}{5}\)

Answer 1

The Correct Option is C

The equation of diagonal OE \(\vec{r}\) = 0+λ(3\(\hat{i}\)+4\(\hat{j}\)+5\(\hat{k}\))

 equation of edge GD 
\(\vec{r}\) = 4\(\hat{j}\) + \(\mu\hat{k}\) 
shortest distance = |projection of 4\(\hat{i}\) on (3\(\hat{j}\) - 4\(\hat{i}\))| 
= \(\frac{12}{\sqrt{9+16}}\) = \(\frac{12}{5}\)

So, the correct answer is (C): \(\frac {12}{5}\)

Answer 2

Shortest Distance Calculation 

Let the rectangular parallelepiped be defined by the vectors a = 3i, b = 4j, and c = 5k. The diagonal OP is given by the vector OP = 3i + 4j + 5k.

Let the edge parallel to the z-axis be denoted by the line segment AB. Since it is not passing through O or P, and it is parallel to the z-axis, we can assume A has coordinates (3, 0, 0) and B has coordinates (3, 0, 5) or another set of coordinates representing a parallel edge. Let’s choose another edge. For example, consider the edge with x = 3, y = 4. So A = (3, 4, 0) and B = (3, 4, 5), which is same as P. We need an edge not passing through P.

Consider the edge parallel to the z-axis passing through (3,0,0). The equation of this line is given by: x = 3, y = 0.

The shortest distance between two skew lines with direction vectors b₁ and b₂, and points on the lines a₁ and a₂ respectively, is given by:

S.D. = $\frac{|(a_2 - a_1) \cdot (b_1 \times b_2)|}{|b_1 \times b_2|}$.

Here, a₁ = (0, 0, 0), b₁ = (3, 4, 5), a₂ = (3, 0, 0), and b₂ = (0, 0, 1).

b₁ × b₂ = $\begin{vmatrix} i & j & k \\ 3 & 4 & 5 \\ 0 & 0 & 1 \end{vmatrix}$ = 4i − 3j.

|b₁ × b₂| = $\sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = 5$.

(a₂ − a₁) = (3, 0, 0) − (0, 0, 0) = (3, 0, 0).

(a₂ − a₁) · (b₁ × b₂) = (3, 0, 0) · (4, −3, 0) = 12.

S.D. = $\frac{12}{5}$