A rectangular parallelepiped with edges along x, y, z axis has length 3, 4, 5 respectively. Fiind the shortest distance of the body diagonal from one of the edges parallel to z-axis which is skew to the diagonal
A rectangular parallelepiped with edges along x, y, z axis has length 3, 4, 5 respectively. Fiind the shortest distance of the body diagonal from one of the edges parallel to z-axis which is skew to the diagonal
- \(\frac{16}{5}\)
- \(\frac{15}{\sqrt{34}}\)
- \(\frac{12}{5}\)
- \(\frac{9}{5}\)
The Correct Option is C
Solution and Explanation
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): \(\frac {12}{5}\)
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Three Dimensional Geometry
Mathematically, Geometry is one of the most important topics. The concepts of Geometry are derived w.r.t. the planes. So, Geometry is divided into three major categories based on its dimensions which are one-dimensional geometry, two-dimensional geometry, and three-dimensional geometry.
Direction Cosines and Direction Ratios of Line:
Consider a line L that is passing through the three-dimensional plane. Now, x,y and z are the axes of the plane and α,β, and γ are the three angles the line makes with these axes. These are commonly known as the direction angles of the plane. So, appropriately, we can say that cosα, cosβ, and cosγ are the direction cosines of the given line L.
