Problem 23 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If \(\vec{a}, \vec{b}, \vec{c}\) are non-coplanar vectors and \(\lambda\) is a real number, then the vectors \(\bar{a}+2 \bar{b}+3 \bar{c}, \lambda \bar{b}+4 \bar{c}\) and \((2 \lambda-1) \bar{c}\) are non coplanar for (a) no value of \(\lambda\) (b) all except one value of \(\lambda\) (c) all except two values of \(\lambda\) (d) all values of \(\lambda\)

Step-by-Step Solution

Answer

(c) all except two values of \(\lambda\)

Step 1: Understanding Non-coplanar Vectors

Vectors are non-coplanar if they do not lie in the same plane, which means the scalar triple product of these vectors is non-zero.

Step 2: Define Given Vectors

Given the vectors: \\[\vec{v_1} = \vec{a} + 2\vec{b} + 3\vec{c}, \quad \vec{v_2} = \lambda \vec{b} + 4\vec{c}, \quad \vec{v_3} = (2\lambda - 1)\vec{c}\]

Step 3: Scalar Triple Product

To check for non-coplanarity, compute the scalar triple product: \\(\vec{v_1} \cdot (\vec{v_2} \times \vec{v_3})\).

Step 4: Cross Product of \(\vec{v_2}\) and \(\vec{v_3}\)

Calculate the cross product \(\vec{v_2} \times \vec{v_3}\): \\[\vec{v_2} \times \vec{v_3} = (\lambda \vec{b} + 4\vec{c}) \times ((2\lambda-1)\vec{c}) = \lambda(2\lambda - 1)(\vec{b} \times \vec{c})\] \since \(\vec{c} \times \vec{c} = \vec{0}\).

Step 5: Calculate Scalar Triple Product

Calculate \(\vec{v_1} \cdot (\vec{v_2} \times \vec{v_3})\): \\[(\vec{a} + 2\vec{b} + 3\vec{c}) \cdot (\lambda(2\lambda - 1)(\vec{b} \times \vec{c})) = \lambda(2\lambda - 1)[\vec{a} \cdot (\vec{b} \times \vec{c})]\] \Given \(\vec{a}, \vec{b}, \vec{c}\) are non-coplanar, \((\vec{a} \cdot (\vec{b} \times \vec{c})) eq 0\).

Step 6: Solve for \(\lambda\)

The scalar triple product is zero when \\[\lambda (2\lambda - 1) = 0\] \Thus, \(\lambda = 0\) or \(2\lambda - 1 = 0\) leading to \(\lambda = \frac{1}{2}\).

Step 7: Conclusion on Non-coplanarity

The vectors are coplanar (scalar triple product is zero) for \(\lambda = 0\) and \(\lambda = \frac{1}{2}\). Otherwise, they are non-coplanar.

Key Concepts

Non-coplanar vectors

In the world of vectors, non-coplanar vectors are those which do not lie on the same plane simultaneously. This property is crucial in determining the spatial arrangement. To check whether three vectors are non-coplanar, we need to compute their scalar triple product. If this product is non-zero, the vectors are non-coplanar. Here,

• Non-coplanar vectors help determine volume in three dimensions.
• They can be visualized as a three-dimensional space that doesn’t flatten into a plane.

This quality is important while solving exercises involving three-dimensional geometry, as it affects equations and vector operations.

Scalar triple product

The scalar triple product is a mathematical tool used to determine if vectors are coplanar or non-coplanar. It involves three vectors: \(\vec{a}, \vec{b}, \vec{c}\). The formula is \(\vec{a} \cdot (\vec{b} \times \vec{c})\), which combines the dot product and the cross product. Since the scalar triple product results in a scalar (a single number), it acts like a confirmation test:

• If non-zero, vectors are non-coplanar.
• If zero, vectors are coplanar.

This scalar helps in calculating volumes defined by vectors, and serves as a determinant of spatial arrangement in geometry. It's key to note that the operation respects vector properties and commutative laws specific to dot and cross products.

Cross product

The cross product, noted as \( \vec{a} \times \vec{b} \), is a vector operation that yields a vector perpendicular to both \(\vec{a}\) and \(\vec{b}\). It's a core piece in the puzzle of non-coplanarity:

• The magnitude of this vector represents the area of the parallelogram spanned by \( \vec{a} \) and \( \vec{b} \).
• The direction follows the right-hand rule, ensuring orthogonality.

When calculating the scalar triple product, the cross product is an intermediary step. By evaluating \( \vec{b} \times \vec{c} \), we define a new vector that is crucial in analyzing spatial relations with a third vector through the dot product.

Non-coplanarity conditions

To understand the conditions for non-coplanarity, we consider the scalar triple product. In this exercise, it has been shown that non-coplanarity is characterized by certain values of \(\lambda\) where the product is non-zero. Specifically:

• When \( \lambda eq 0 \) and \( \lambda eq \frac{1}{2} \), vectors are non-coplanar.
• When \( \lambda \) equals these values, the vectors become coplanar.

The condition for coplanarity can directly influence calculations and solutions involving those vectors. Such understanding assists in determining feasible solutions to geometric problems or in physical computations where spatial understanding is pivotal.