Problem 28 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The perpendicular distance of a corner of a unit cube form a diagonal not passing through it is (A) \(\sqrt{6}\) (B) \(\frac{\sqrt{6}}{3}\) (C) \(\frac{3}{\sqrt{6}}\) (D) none of these

Step-by-Step Solution

Answer

The correct answer is (B) \(\frac{\sqrt{6}}{3}\).

Step 1: Understanding the Problem

We have a unit cube where each side has length 1. We're asked to find the perpendicular distance from a corner to a space diagonal that does not pass through that corner.

Step 2: Identifying the Space Diagonal

A space diagonal of the cube connects opposite corners. Let's assume the cube is positioned with vertices at (0,0,0) and (1,1,1). The diagonal we're considering does not pass through the origin (0,0,0) but extends from the vertex (1,1,1) to (0,0,0).

Step 3: Finding the Equation of the Diagonal

The diagonal is a line segment from (0,0,0) to (1,1,1). The vector form of this diagonal is given by \((t, t, t)\) where \(t\) ranges from 0 to 1. This means any point on the diagonal can be represented as (t,t,t).

Step 4: Position of the Corner

Since we are looking for a corner that the diagonal does not pass through, let's consider the point (1,0,0). This does not lie on our chosen diagonal.

Step 5: Calculating the Distance Formula

The distance formula between a point and a line in vector form can be given by \( \frac{\| \vec{PQ} \times \vec{d} \|}{\|\vec{d}\|} \), where \(\vec{PQ}\) is a vector from point P to the line, \(\vec{d}\) is the direction vector of the line, and \(\times\) represents the cross product.

Step 6: Defining the Vectors

For point Q (1,0,0), we have \(\vec{PQ} = \langle 1-t, -t, -t \rangle\) and \(\vec{d} = \langle 1, 1, 1 \rangle\).

Step 7: Calculating the Cross Product

We find the cross product \(\vec{PQ} \times \vec{d}\):\[(i, j, k) \begin{vmatrix} 1-t & -t & -t \ 1 & 1 & 1 \end{vmatrix} = \langle 0, -1, 1 \rangle\]

Step 8: Finding the Magnitudes

The magnitude of the cross product vector is \(\sqrt{0^2 + (-1)^2 + 1^2} = \sqrt{2}\), and the magnitude of \(\vec{d}\) is \(\sqrt{1^2 + 1^2 + 1^2} = \sqrt{3}\).

Step 9: Calculating the Distance

The perpendicular distance from (1,0,0) to the diagonal is given by \( \frac{\sqrt{2}}{\sqrt{3}} = \frac{\sqrt{6}}{3} \).

Step 10: Conclusion

Thus, the perpendicular distance from the corner (1,0,0) of the unit cube to the diagonal is \(\frac{\sqrt{6}}{3}\). Therefore, the correct option is (B).

Key Concepts

unit cube

A unit cube is a three-dimensional shape where each side is exactly one unit long. This makes all edges of the cube equal in length. It's a basic building block in geometry and helps in understanding more complex structures. When you imagine a small cube that you could hold between your fingers, that's a unit cube.

Here are a few essential features of a unit cube:

• Each face of the cube is a square with an area of 1 square unit.
• The cube has 6 faces, 12 edges, and 8 vertices.
• All the angles in a unit cube are right angles, which means 90 degrees.
• It's symmetrical, meaning it looks the same from all directions.

Understanding the unit cube aids in visualizing problems involving three-dimensional spaces. It forms the basis of calculating volumes, surface areas, and distances in geometry.

space diagonal

In geometry, a space diagonal is a line that stretches from one corner of a cube to its opposite corner. It passes through the interior of the cube, unlike the face diagonal, which only passes across the surface. This diagonal is longer than the sides of the cube because it spans the entire three-dimensional space within.

Key aspects of a space diagonal in a unit cube:

• To find the length of a space diagonal in a unit cube, you use the formula for the diagonal of a cube: \( ext{length} = \sqrt{3} \times ext{side length} \). For a unit cube, this equals \( \sqrt{3} \).
• Space diagonals help in understanding distances in three-dimensional geometry.
• Each cube has four space diagonals, connecting opposite vertices.

In problems like the one we're examining, understanding the concept of a space diagonal is crucial for calculating the perpendicular distance from points within the cube to these diagonals.

perpendicular distance

The perpendicular distance is the shortest distance from a point to a line or a surface. In geometry, it's akin to dropping an imaginary perpendicular line from the point to intersect the line at a 90-degree angle.

Here's how to think about perpendicular distance in the context of the given problem:

• The problem involves finding the perpendicular distance from one corner of a unit cube to a space diagonal.
• To find this distance accurately, we use the formula that involves the cross product of vectors, which helps in determining how far off-set the point is from the diagonal line.
• Using the formula: \( \frac{\| \vec{PQ} \times \vec{d} \|}{\|\vec{d}\|} \), we can calculate this shortest distance.

Understanding this concept is crucial for solving problems involving distances in three-dimensional geometry, providing a practical application of mathematical formulas in space perception.