Let Q be the cube with the set of vertices {(x1, x2, x3) ∈ R3: x1, x2, x3 ∈ {0,1}}. Let F be the set of all twelve lines containing the diagonals of the six faces of cube Q. Let S be the set of all four lines containing the main diagonals of the cube Q; for instance, the line passing through the vertices (0,0,0) and (1,1,1) is in S. For lines l1 and l2, let d(l1,l2) denote the shortest distance between them. Then the maximum value of d(l1,l2) as l1 varies over f and l2 varies over S, is
Let Q be the cube with the set of vertices {(x1, x2, x3) ∈ R3: x1, x2, x3 ∈ {0,1}}. Let F be the set of all twelve lines containing the diagonals of the six faces of cube Q. Let S be the set of all four lines containing the main diagonals of the cube Q; for instance, the line passing through the vertices (0,0,0) and (1,1,1) is in S. For lines l1 and l2, let d(l1,l2) denote the shortest distance between them. Then the maximum value of d(l1,l2) as l1 varies over f and l2 varies over S, is
- \(\frac{1}{\sqrt6}\)
- \(\frac{1}{\sqrt8}\)
- \(\frac{1}{\sqrt3}\)
- \(\frac{1}{\sqrt{12}}\)
The Correct Option is A
Solution and Explanation
The correct option is (A):
The equation of the OD line is
\(\vec{r_1}=\vec{0}+\lambda(\hat{i}+\hat{j})\)
equation of diagonal BE is
\(\vec{r_1}=\hat{j}+\alpha(\hat{i}-\hat{j}+\hat{k})\)
S.D = \(\left | \frac{\hat{j}.(\hat{i}-\hat{j}+\hat{k})}{\sqrt{6}} \right |=\frac{1}{\sqrt6}\)
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Concepts Used:
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.
