© All Rights reserved @ LearnWithDash
Step 1: Understand the Problem
We have a rectangular parallelepiped (a 3D box) with one vertex at the origin O(0,0,0) and the opposite vertex at P(3,4,5). The edges of this parallelepiped are aligned with the coordinate axes and have lengths 3, 4, and 5 along the x, y, and z axes, respectively. We want to determine the shortest distance between the diagonal OP and any edge of the parallelepiped that is parallel to the z-axis and does not pass through O or P.
Step 2: Identify the Relevant Edges Parallel to the z-Axis
Apart from the edges passing through O or P, there are two edges parallel to the z-axis:
• The line segment joining (3,0,0) to (3,0,5).
• The line segment joining (0,4,0) to (0,4,5).
By symmetry, each of these will give the same shortest distance to diagonal OP, so it suffices to analyze just one (for instance, the line through (3,0,z)).
Step 3: Express Both Lines in Vector Form
1) Line OP
• A point on the line: O(0,0,0).
• Direction vector:
$ \vec{OP} = (3,4,5). $
• Parametric/vector equation:
$ (x,y,z) = (0,0,0) + t(3,4,5), \; t \in \mathbb{R}. $
2) The chosen edge (through x=3, y=0)
• A point on the line: A(3,0,0).
• Direction vector (along z-axis):
$ \vec{k} = (0,0,1). $
• Parametric/vector equation:
$ (x,y,z) = (3,0,0) + s(0,0,1) = (3,0,s), \; s \in \mathbb{R}. $
Step 4: Recall the Formula for the Shortest Distance Between Two Skew Lines
For two skew lines given by
$ \vec{r}_1 = \vec{a}_1 + \lambda \vec{d}_1 $ and
$ \vec{r}_2 = \vec{a}_2 + \mu \vec{d}_2, $
the shortest distance between them is
$ \displaystyle d = \frac{|(\vec{a}_2 - \vec{a}_1)\cdot(\vec{d}_1 \times \vec{d}_2)|}{|\vec{d}_1 \times \vec{d}_2|}. $
Step 5: Identify the Vectors in the Formula
• $ \vec{a}_1 = (0,0,0) $ (point on line OP)
• $ \vec{d}_1 = (3,4,5) $ (direction of OP)
• $ \vec{a}_2 = (3,0,0) $ (point on the chosen z-axis edge)
• $ \vec{d}_2 = (0,0,1) $ (direction of the z-axis edge)
Step 6: Compute $ \vec{a}_2 - \vec{a}_1 $ and the Cross Product $ \vec{d}_1 \times \vec{d}_2 $
1) $ \vec{a}_2 - \vec{a}_1 = (3,0,0) - (0,0,0) = (3,0,0). $
2) $ \vec{d}_1 \times \vec{d}_2 =
\begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
3 & 4 & 5 \\
0 & 0 & 1
\end{vmatrix}
= (4 \cdot 1 - 5 \cdot 0)\mathbf{i} - (3 \cdot 1 - 5 \cdot 0)\mathbf{j} + (3 \cdot 0 - 4 \cdot 0)\mathbf{k}
= (4, -3, 0). $
Step 7: Calculate the Required Dot Product and Magnitude
• Dot product in the numerator:
$ (\vec{a}_2 - \vec{a}_1)\cdot(\vec{d}_1 \times \vec{d}_2) = (3,0,0)\cdot(4, -3, 0) = 3 \times 4 + 0 \times (-3) + 0 \times 0 = 12. $
• Magnitude of the cross product in the denominator:
$ |\vec{d}_1 \times \vec{d}_2| = \sqrt{4^2 + (-3)^2 + 0^2} = \sqrt{16 + 9} = \sqrt{25} = 5. $
Step 8: Determine the Shortest Distance
Putting these together in the formula for shortest distance:
$ d = \frac{|12|}{5} = \frac{12}{5}. $
Final Answer
The shortest distance between the diagonal OP of the parallelepiped and the chosen edge parallel to the z-axis is
$ \displaystyle \frac{12}{5}. $
