Problem 30 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If the four points \(a, b, c, d\) are coplanar, then \([b c d]+[c a d]+[a b d]=\) (A) 0 (B) 1 (C) \(-1\) (D) \([a b c]\)

Step-by-Step Solution

Answer

The answer is (A) 0.

Step 1: Understanding Coplanar Points

Four points \(a, b, c,\) and \(d\) are coplanar if any volume they encompass is zero. In 3D space, the determinant of the matrix formed by vectors \(\overrightarrow{ab}, \overrightarrow{ac},\) and \(\overrightarrow{ad}\) represents the volume of the parallelepiped they form. If the points are coplanar, this volume (or triple product) is zero.

Step 2: Evaluate the Determinants

The symbol \([bcd]\) represents the scalar triple product (or determinant of a matrix) formed by vectors \(\overrightarrow{bc}, \overrightarrow{bd},\) and \(\overrightarrow{cd}\), and similarly for \([cad]\) and \([abd]\). We need to show that \([bcd] + [cad] + [abd] = 0\).

Step 3: General Properties of Determinants

Recall that the sum of determinants of vectors forming a closed loop results in zero. That is, for coplanar points, the sum of determinants involving permutations of three vectors should result in zero due to how determinants behave under linear transformations and coplanarity.

Step 4: Sum the Determinants

Given the coplanarity condition, applying the properties of determinants (or vector products) linked to four coplanar points: \([bcd] + [cad] + [abd] = 0\). This exploits the fact that adding vectors representing the same plane in different orders naturally sums to zero.

Key Concepts

Scalar Triple Product

The scalar triple product is a fundamental concept in 3D geometry, primarily used to determine the volume of a parallelepiped formed by three vectors. It is calculated as the dot product of one vector with the cross product of the other two vectors. If you have vectors \(\mathbf{a}, \mathbf{b}, \text{ and } \mathbf{c}\), the scalar triple product is \(\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})\). This expression can also be presented as a determinant:

• \([\mathbf{a} \mathbf{b} \mathbf{c}] = \begin{vmatrix} a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \ c_1 & c_2 & c_3 \end{vmatrix}\)

Being a scalar value, it is crucial for establishing the coplanarity of points. When the scalar triple product equals zero, it indicates that the vectors, and hence the points they define, are coplanar. This is because a volume of zero in a 3-dimensional space (which the product represents) implies that the vectors lie in the same plane, resulting in no 'height' perpendicular to the plane.

Determinants in 3D Geometry

Determinants play a pivotal role in 3D geometry by facilitating the calculation of various geometric properties. A determinant, in the context of vectors forming a matrix, measures a volume-like entity or area depending on the dimension. Specifically, when using determinants of 3x3 matrices, we assess properties related to three-dimensional space. The determinants help in:

• Calculating the volume of parallelepipeds using the scalar triple product.
• Determining if vectors (and their resulting points) are coplanar.

For a matrix formed by vectors \(\mathbf{a}, \mathbf{b}, \text{ and } \mathbf{c}\), the determinant, expressed as:\[\begin{vmatrix} a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \ c_1 & c_2 & c_3 \end{vmatrix}\]is the volumetric extension of how these vectors span the space. If the determinant value is zero, it implies the vectors fail to cover a 3D space, indicating planarity, as they lie on a single plane.

Properties of Determinants

Understanding the properties of determinants is crucial when addressing coplanarity problems. Determinants possess several essential properties, some of which are directly applicable in proving coplanarity:

• Linear Dependence: If the rows or columns of a matrix are linearly dependent, the determinant is zero. This reflects directly when considering vectors, as dependent vectors lie in the same plane.
• Operational Effects: The determinant of a matrix changes sign when any two rows or columns are swapped.
• Additivity: Adding a multiple of one row to another does not change the determinant's value, making it stable under such operations.
• Effect of Scaling: If any row or column is scaled by a factor, the whole determinant will change by that factor.

Applying these properties in 3D geometry helps confirm the zero-sum scenario of coplanarity. Especially in exercises where calculating the sum of several determinant values reveals if the corresponding vectors are planar, as in the provided exercise. The simplicity of linear transformations within determinants makes them invaluable tools in higher-dimensional geometry assessments.