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

Question

If the four points \(a, b, c, d\) are coplanar, then \(\left[\begin{array}{ll}b c d\end{array}\right]+\left[\begin{array}{ll}c & a & d\end{array}\right]+\left[\begin{array}{ll}a & b & d\end{array}\right]=\) (A) 0 (B) 1 (C) \(-1\) (D) \(\left[\begin{array}{lll}a & b & c\end{array}\right]\)

Step-by-Step Solution

Answer

(A) 0

Step 1: Understanding the Determinant and Coplanar Vectors

If four points \(a, b, c, d\) are coplanar, any vector formed from these points can be expressed as a linear combination of the others. The associated vectors \( \overrightarrow{AB}, \overrightarrow{AC}, \overrightarrow{AD} \) must be linearly dependent.

Step 2: Determinant Properties of Coplanar Vectors

For coplanar vectors \(u, v, w\), the determinant composed of these vectors \[Det(u, v, w) = \left| \begin{array}{ccc} u_1 & u_2 & u_3 \ v_1 & v_2 & v_3 \ w_1 & w_2 & w_3 \end{array} \right| = 0\] must be zero if they are also constructed from coplanar points.

Step 3: Identifying the Determinants in the Expression

The expression \[\left[ \begin{array}{lll} b & c & d \end{array} \right] + \left[ \begin{array}{lll} c & a & d \end{array} \right] + \left[ \begin{array}{lll} a & b & d \end{array} \right]\] calls for three determinants. We recognize potential dependency when any combination results in zero.

Step 4: Measure Determinant Contribution to Coplanarity

Calculate the sum of these determinants. Relying on the properties of determinants, every linear related set from coplanar points may display a summation equalling the fourth point.'s contribution.

Step 5: Solution Verification

Each determinant includes the extraneous term, omitting it results in full coplanarity, verifying the function of determinants via expansion or corevice calculations. When vectors are coplanar, the discussed determinant sum equates to zero from configuration alignment.

Key Concepts

Determinant Properties

Determinants play a crucial role in determining whether vectors are coplanar. When working with vectors formed by points in space, the determinant of a matrix constructed from these vectors can tell us if they lie in the same plane. This is because, for three vectors, if the determinant of their corresponding 3x3 matrix is zero, the vectors are linearly dependent and lie along the same plane (coplanar).

In mathematical terms, consider vectors \(\mathbf{u}, \mathbf{v}, \mathbf{w}\). The determinant of these vectors is given by:\[\text{Det}(\mathbf{u}, \mathbf{v}, \mathbf{w}) = \left| \begin{array}{ccc} u_1 & u_2 & u_3 \ v_1 & v_2 & v_3 \ w_1 & w_2 & w_3 \end{array} \right|\]To check coplanarity, if this determinant equals zero, the vectors lie on a single plane. This property simplifies the calculation and makes it straightforward to check if vectors or points in 3D space are coplanar.

Linear Dependence

Understanding linear dependence is essential when dealing with coplanar vectors. Linear dependence occurs when some vectors in a set can be expressed as linear combinations of others within the set. In a matrix composed of vectors, if at least one vector is a combination of the others, they do not span the entire space, but are constrained to a lower dimension.

For vectors to be coplanar, they must be linearly dependent. If vectors \(\mathbf{u}, \mathbf{v}, \mathbf{w}\) are such that \(c_1 \mathbf{u} + c_2 \mathbf{v} + c_3 \mathbf{w} = 0\) for constants not all zero, these vectors are linearly dependent. This relation signifies they reside in the same plane. Thus, recognizing linear dependence through determinants or vector equations is a key step in identifying coplanarity.

In real-world applications, understanding this concept helps solve geometric problems and optimize computations involving multidimensional data.

Vectors and Geometry

Vectors are fundamental in exploring and understanding geometry. They provide both direction and magnitude, making them an indispensable tool in many areas of mathematics and physics. In geometry, vectors are often used to represent points, lines, and planes in a coordinate system.

The relationship between vectors and their geometric representation is significant when determining properties like parallelism, perpendicularity, and importantly, coplanarity. To work with vectors in a geometric context, one often uses operations like vector addition, scalar multiplication, dot product, and cross product.

• Vector Addition: Combining vectors to find resultant vectors, often used in navigation and physics.
• Scalar Multiplication: Scaling vectors to alter magnitude, keeping direction constant.
• Dot Product: Calculating to find angles or test perpendicularity between vectors.
• Cross Product: Useful for finding normal vectors to planes, crucial in determining coplanarity.

These operations allow one to describe and manipulate geometric entities conveniently and form the basis of further studies in vector calculus and physics.