Problem 82 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Statement 1: If the points \((1,2,2),(2,1,2)\) and \((2,2, z)\) and \((1,1,1)\) are coplanar, then \(z=2\). Statement 2: If the 4 points \(P, Q, R\) and \(S\) are coplanar, then the volume of the tetrahedron \(P Q R S\) is 0 . [Online May 12, 2012] (a) Statement 1 is false,, Statement 2 is true. (b) Statement 1 is true, Statement 2 is false. (c) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation of Statement 1 . (d) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation of Statement 1 .

Step-by-Step Solution

Answer

(a) Statement 1 is false, Statement 2 is true.

Step 1: Understanding Coplanarity

For four points to be coplanar, the volume of the tetrahedron formed by these points must be zero. The volume can be calculated using the scalar triple product. If this value is zero, the points are coplanar.

Step 2: Setting Up the Scalar Triple Product

Use the points given: \(A(1,2,2)\), \(B(2,1,2)\), \(C(2,2,z)\), and \(D(1,1,1)\). We consider vector \(\overrightarrow{AB} = (2-1, 1-2, 2-2) = (1, -1, 0)\), \(\overrightarrow{AC} = (2-1, 2-2, z-2) = (1, 0, z-2)\), and \(\overrightarrow{AD} = (1-1, 1-2, 1-2) = (0, -1, -1)\).

Step 3: Calculate the Scalar Triple Product

The scalar triple product is \(((\overrightarrow{AB} \times \overrightarrow{AC}) \cdot \overrightarrow{AD})\). First find \(\overrightarrow{AB} \times \overrightarrow{AC} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 1 & -1 & 0 \ 1 & 0 & z-2 \end{vmatrix}\). Calculate the determinant: \(((-1)(z-2) - (0)(0))\mathbf{i} - ((0)(1) - (0)(z-2))\mathbf{j} + ((1)(0) - (-1)(1))\mathbf{k}\). This results in \((-z+2)\mathbf{i} + 1\mathbf{k}\).

Step 4: Dot Product with \(\overrightarrow{AD}\)

Take the dot product of \((-z+2)\mathbf{i} + 1\mathbf{k}\) with \(\overrightarrow{AD} = (0, -1, -1)\), giving \(((0)(-z+2) + (-1)(0) + 1(-1)) = -1\).

Step 5: Conclusion from Scalar Triple Product

Since the scalar triple product is \(-1\), which is not zero, the points are not coplanar, meaning that Statement 1 is false.

Step 6: Checking Statement 2

Statement 2 correctly states that if four points are coplanar, the volume of the tetrahedron they form is zero. This is a true statement by definition.

Step 7: Determining the Correct Answer

Since Statement 1 is false and Statement 2 is true, the correct answer is option (a).

Key Concepts

Coplanarity

Coplanarity in the context of 3D geometry refers to the condition where a set of points lie on the same plane. For four points to be coplanar, the volume of the tetrahedron formed by these points must be zero. This is a key idea that helps us to determine if four given points are on the same plane or not.
One of the powerful tools to verify coplanarity is the Scalar Triple Product. It provides a mathematical test to check if vectors formed by these points are coplanar:

• Define vectors with the given points.
• Form the Scalar Triple Product using these vectors.
• If the product is zero, the points are coplanar.

In our exercise, the Scalar Triple Product was found to be non-zero, proving that the points were not coplanar. This technique is an essential part of vector calculus that simplifies geometrical validations in 3D space.

Volume of Tetrahedron

The volume of a tetrahedron in 3D geometry can be understood through vectors and the scalar triple product. A tetrahedron is a three-dimensional figure composed of four triangular faces. The volume is zero if all four points (vertices) lie on the same plane.
The volume can be calculated using the scalar triple product of vectors originating from one vertex to the others. Mathematically, it is given by:
\[ V = \frac{1}{6} |(\overrightarrow{AB} \times \overrightarrow{AC}) \cdot \overrightarrow{AD}| \]
where the vectors represent the sides of the tetrahedron emanating from one vertex. If this determinant (part of the scalar triple product calculation) equals zero, then the volume is zero, confirming coplanarity.
In the exercise, although we setup the vectors for volume determination, the non-zero scalar triple product indicated a non-zero volume, affirming non-coplanarity.

Vectors in 3D Geometry

Vectors are essential tools in 3D geometry as they help in representing quantities having both magnitude and direction. In problems involving spatial geometry, vectors simplify calculations and provide a method to handle multiple dimensions efficiently.
One of their important applications is determining geometric properties like the coplanarity of points or the volume of shapes like tetrahedrons. When dealing with points in 3D, vectors help create meaningful representations, such as:

• Connecting points to form line segments.
• Performing vector operations like addition, scalar multiplication, and particularly, the cross and dot products needed for volume and coplanarity checks.

Vectors also allow for the transformation and representation of shapes in different coordinate systems, which provides flexibility in solving complex problems. Understanding how vectors work is an invaluable skill when delving into 3D geometry tasks.