Problem 45 - A Complete Resource Book in Mathematics for JEE Main

Question

A parallelopiped is formed by planes drawn through the points \((2,3,5)\) and \((5,9,7)\), parallel to the coordi- nate planes. The length of a diagonal of the parallelopiped is: \([\mathbf{2 0 0 2}]\) (A) 7 unit (B) \(\sqrt{38}\) unit (C) \(\sqrt{155}\) unit (D) none of these

Step-by-Step Solution

Answer

The length of the diagonal is 7 units (Option A).

Step 1: Understand the Parallelepiped

A parallelepiped is a six-faced figure (also called a box) where each face is a parallelogram. Here, planes pass through two given points \( (2,3,5) \) and \( (5,9,7) \), and are parallel to coordinate planes. This means the parallelepiped's edges are parallel to the x, y, and z axes.

Step 2: Determine the Dimensions of the Parallelepiped

The dimensions of the parallelepiped can be found by calculating the differences along each axis between the two points: - Length along x-axis: \( 5 - 2 = 3 \)- Length along y-axis: \( 9 - 3 = 6 \)- Length along z-axis: \( 7 - 5 = 2 \)Thus, the parallelepiped has dimensions 3, 6, and 2.

Step 3: Calculate the Diagonal Using the Distance Formula

To find the length of the diagonal of the parallelepiped, use the 3-dimensional distance formula: \[\text{Diagonal} = \sqrt{(\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2 }\]Substituting the dimensions:\[\text{Diagonal} = \sqrt{3^2 + 6^2 + 2^2 } = \sqrt{9 + 36 + 4 } = \sqrt{49}\]

Step 4: Simplify the Diagonal Length

Simplify \( \sqrt{49} \) to find the length of the diagonal:\[\sqrt{49} = 7\]Thus, the length of the diagonal is 7 units.

Key Concepts

Parallelepiped

A parallelepiped is a 3-dimensional shape similar to a rectangle but with parallelogram faces, not rectangles. Imagine a box that can lean over on one side, making it less boxy and more slanted.
This shape has six faces, and each face is a parallelogram. A distinct feature is that opposite faces are parallel. Like a cube, a parallelepiped also has vertices, edges, and faces, but it's generally more versatile in shape.
In our exercise, this shape is formed by drawing planes through the given points, parallel to the coordinate planes. This results in a structure where the edges line up neatly with the x, y, and z axes. This alignment allows us to easily calculate dimensions because the lengths along each axis can be found simply by subtracting coordinates.

Distance Formula

The distance formula is a crucial tool in geometry. It helps calculate the precise distance between two points in space. For two points in 3D space, \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\), the distance formula is used as follows:

• \( \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2 } \)
  This formula extends Pythagoras' theorem into three dimensions. Imagine forming a right-angled triangle by connecting these points with the axes.
• In the exercise, the parallelepiped’s diagonal length is calculated by applying this formula to determine the straight-line distance between the opposite corners. The sides of the parallelepiped act as the triangle’s legs, while the diagonal is the hypotenuse.

Using this formula, you can find distances in any 3-dimensional object, not just parallelepipeds.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, uses the Cartesian coordinate system to describe geometric figures using algebraic equations.

• It allows the calculation of areas, distances, and other facets of geometry directly from coordinates. This method makes deriving lengths, angles, and other geometric properties systematic and accessible.

The Cartesian coordinate system is a grid that allows you to specify unique positions with coordinates \( (x, y, z) \). By working in this system, problems involving shapes like parallelepipeds become easy to solve because you can handle them numerically rather than just visually or descriptively. The transformation of geometric problems to algebraic ones makes it easier to solve complex geometrical problems. This is why coordinate geometry serves as a strong base for working with 3-dimensional shapes.