GATE CS 2021 Set 2 — Question 3
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General Aptitude → Quantitative Aptitude → Mensuration & Geometry
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Question
If is the angle, in degrees, between the longest diagonal of the cube and any one of the edges of the cube, then, =
Correct answer
(B) 1√(3)
Solution
To find the angle between the longest diagonal and an edge of a cube, we can use vector analysis.
1.Let's place the cube in a 3D Cartesian coordinate system with one vertex at the origin (0, 0, 0) and its edges aligned with the x, y, and z axes. Let the side length of the cube be 'a'.
2.The longest diagonal (or space diagonal) connects opposite vertices. Let's consider the diagonal from the origin (0, 0, 0) to the vertex (a, a, a). The vector representing this diagonal is:
3.The magnitude of the diagonal vector is:
4.Now, let's consider one of the edges connected to the origin. Due to the symmetry of the cube, the angle will be the same for any edge. Let's choose the edge along the x-axis. The vector representing this edge is:
5.The magnitude of the edge vector is:
6.The angle between two vectors and is given by the dot product formula:
7.First, calculate the dot product:
8.Now, solve for :
Thus, the correct option is (B).Continue learning with Success Tracker
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