Problem 13 - A Complete Resource Book in Mathematics for JEE Main

Question

The lines \(\mathbf{r}=\mathbf{a}+\lambda(\mathbf{b} \times \mathbf{c})\) and \(\mathbf{r}=\mathbf{b}+\mu(\mathbf{c} \times \mathbf{a})\) will intersect if (A) \(\mathbf{a} \times \mathbf{c}=\mathbf{b} \times \mathbf{c}\) (B) \(\mathbf{a} \cdot \mathbf{c}=\mathbf{b} \cdot \mathbf{c}\) (C) \(\mathbf{b} \times \mathbf{a}=\mathbf{c} \times \mathbf{a}\) (D) none of these

Step-by-Step Solution

Answer

The lines intersect if \( \mathbf{a} \times \mathbf{c} = \mathbf{b} \times \mathbf{c} \), thus option (A).

Step 1: Equations of Intersection

To determine if the lines intersect, set their position vectors equal. From the lines \( \mathbf{r} = \mathbf{a} + \lambda(\mathbf{b} \times \mathbf{c}) \) and \( \mathbf{r} = \mathbf{b} + \mu(\mathbf{c} \times \mathbf{a}) \), equate: \[ \mathbf{a} + \lambda(\mathbf{b} \times \mathbf{c}) = \mathbf{b} + \mu(\mathbf{c} \times \mathbf{a}) \].

Step 2: Rearrange the Equation

Rearrange the equation from Step 1 to find \( \mathbf{a} - \mathbf{b} = \mu(\mathbf{c} \times \mathbf{a}) - \lambda(\mathbf{b} \times \mathbf{c}) \). This can be rewritten as: \[ \mathbf{a} - \mathbf{b} = \mu(\mathbf{c} \times \mathbf{a}) - \lambda(\mathbf{b} \times \mathbf{c}) \].

Step 3: Apply Vector Identity

Note that the vector identity \( \mathbf{b} \times \mathbf{c} = -\mathbf{c} \times \mathbf{b} \) may simplify our equation. If we focus on the vectors that cause zero resulting vectors, we aim that both \( \mathbf{b} \times \mathbf{c} = \mathbf{c} \times \mathbf{a} \), or vice-versa cancel out testing interactions among options.

Step 4: Test \\( \mathbf{a} \times \mathbf{c}=\mathbf{b} \times \mathbf{c} \\)

Use Test Condition (A): Substitute in the equation \[ \mathbf{a} \times \mathbf{c}=\mathbf{b} \times \mathbf{c} \]. If they are equal implying zero magnitude then both \( \lambda(\mathbf{b} \times \mathbf{c}) \) and \( \mu(\mathbf{c} \times \mathbf{a}) \) vanish having aligned \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) vectors.

Step 5: Perform the Intersection Verification

Since \( \mathbf{a} \times \mathbf{c} = \mathbf{b} \times \mathbf{c} \) enables all vector cross-terms aligns to zero equivalence for intersection solutions. Thus, Condition (A) is satisfied providing intersection between the line configuration.

Key Concepts

Intersection of Lines

In vector algebra, the intersection of lines is pivotal when analyzing spaces and trajectories. The concept involves determining if two lines meet at a certain point. To find out if given lines intersect, we equate their vector equations. For example, with lines given by \( \mathbf{r} = \mathbf{a} + \lambda(\mathbf{b} \times \mathbf{c}) \) and \( \mathbf{r} = \mathbf{b} + \mu(\mathbf{c} \times \mathbf{a}) \):
* Equate their position vectors, resulting in a system of equations.
* Rearrange to isolate terms related to direction vectors.
* Solve for parameters such as \( \lambda \) and \( \mu \) to meet a common point.
If possible, a set of parameter values equalizes both equations, indicating intersection. Understanding this concept involves grasping the geometric interpretation of vectors and visualizing how vector arithmetic governs line behaviors in space.

Cross Product

The cross product is a fundamental vector operation that helps us understand spatial relationships between vectors. Formally, the cross product of two vectors \( \mathbf{u} \) and \( \mathbf{v} \) is given by \( \mathbf{u} \times \mathbf{v} \). This operation yields a third vector that is perpendicular to both \( \mathbf{u} \) and \( \mathbf{v} \). Important characteristics:
* The magnitude of the cross product is equal to the area of the parallelogram spanned by the two vectors.
* If the vectors are parallel, their cross product is zero.
* The cross product is anti-commutative, meaning \( \mathbf{u} \times \mathbf{v} = - (\mathbf{v} \times \mathbf{u}) \).
In the context of our problem, vector cross products \( \mathbf{b} \times \mathbf{c} \) and \( \mathbf{c} \times \mathbf{a} \) play a crucial role in determining line directions and verifying intersections.

Vector Equations

Vector equations form the backbone of describing lines in three-dimensional space. A line can be described as the set of all points that satisfy a vector equation such as \( \mathbf{r} = \mathbf{a} + \lambda \mathbf{d} \), where:
* \( \mathbf{r} \) represents a position vector on the line.
* \( \mathbf{a} \) is a fixed point on the line.
* \( \mathbf{d} \) is the direction vector indicating the line's orientation.
* \( \lambda \) is a scalar parameter that travels along the line.
These components work together to map every point on infinite lines with varying direction scales. By manipulating vector equations and modifying parameters, we establish intersection criteria as seen in algebraic intersections or by visual imagery using direction constancy across equations for line collinearity checks.