Problem 13 - A Complete Resource Book in Mathematics for JEE Main 2017

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 correct answer is (A) \( \mathbf{a} \times \mathbf{c} = \mathbf{b} \times \mathbf{c} \).

Step 1: Understanding Line Equations

We begin by understanding the lines given:1. The first line is given by \( \mathbf{r}=\mathbf{a}+\lambda(\mathbf{b} \times \mathbf{c}) \), where \( \mathbf{r} \) is a position vector, and \( \lambda \) is a scalar.2. The second line is \( \mathbf{r}=\mathbf{b}+\mu(\mathbf{c} \times \mathbf{a}) \), where \( \mu \) is another scalar.For the lines to intersect, their position vectors at any given \( \lambda \) and \( \mu \) should be equal.

Step 2: Setting Equations Equal

To solve for intersection points, set the position vectors equal:\[ \mathbf{a} + \lambda(\mathbf{b} \times \mathbf{c}) = \mathbf{b} + \mu(\mathbf{c} \times \mathbf{a}) \]This can be rearranged to:\[ \mathbf{a} - \mathbf{b} = \mu(\mathbf{c} \times \mathbf{a}) - \lambda(\mathbf{b} \times \mathbf{c}) \]

Step 3: Analyzing the Cross Product

The cross-product terms imply that if \( \mathbf{c} \times \mathbf{a} \) is in the direction of \( \mathbf{b} \times \mathbf{c} \), then both sides of the equation can balance out, potentially leading the vectors \( \mathbf{b} \) and \( \mathbf{a} \) to align such that the equation becomes consistent at some pair \( (\lambda, \mu) \).

Step 4: Interpreting Cross Product Condition

The condition for the orthogonality in the cross product forms, noticing the terms, stems from the geometric property of cross products: they are zero for parallel vectors. If \( \mathbf{b} \) and \( \mathbf{a} \) are such that their operations with \( \mathbf{c} \) yield the same direction, then:\( \mathbf{b} \times \mathbf{c} = \mathbf{c} \times \mathbf{a} \)This alternately implies that \( \mathbf{a} \times \mathbf{c} = \mathbf{b} \times \mathbf{c} \) (same direction property with reversed terms).

Step 5: Final Verification and Conclusion

Re-assessing the options, only (A) \( \mathbf{a} \times \mathbf{c} = \mathbf{b} \times \mathbf{c} \) is found to naturally align with the derived equality. Given the properties of cross products across the same plane, this supports the condition aligning with the geometric intersection analysis.

Key Concepts

Cross Product

The cross product is a key vector operation in vector algebra. In simple terms, the cross product of two vectors, say \( \mathbf{u} \) and \( \mathbf{v} \), results in a third vector that is orthogonal (perpendicular) to both. This means if \( \mathbf{u} \times \mathbf{v} = \mathbf{w} \), then \( \mathbf{w} \) makes a 90° angle with both \( \mathbf{u} \) and \( \mathbf{v} \).

• It is calculated using the determinant of a 3x3 matrix formed by the unit vectors and the vectors involved.
• The magnitude of the cross product \( ||\mathbf{u} \times \mathbf{v}|| \) is equal to the area of the parallelogram formed by \( \mathbf{u} \) and \( \mathbf{v} \).

This property is fundamental in determining whether two vectors are parallel. If the cross product is zero, the vectors are parallel. This plays a part in determining the condition for lines represented by vector equations to intersect geometrically.

Geometric Intersection

Geometric intersection refers to the point where two lines cross each other. For vector lines, this happens when their vector equations produce the same vector, meaning the lines share a common point.

• Given lines \( \mathbf{r}_1 = \mathbf{a} + \lambda (\mathbf{b} \times \mathbf{c}) \) and \( \mathbf{r}_2 = \mathbf{b} + \mu (\mathbf{c} \times \mathbf{a}) \), the intersection occurs if there exists a \( \lambda \) and \( \mu \) such that \( \mathbf{r}_1 = \mathbf{r}_2 \).
• The solution involves equating the two vector expressions and solving for the scalars \( \lambda \) and \( \mu \).

Determining intersection is crucial in 3D graphics, physics simulations, and robotics, where path overlap can affect motion analysis and collision detection.

Vector Equation

Vector equations provide a way to express geometric entities like lines and planes in algebraic form using vectors. A typical line equation involves a position vector and a direction vector, scaled by a parameter.

• The line \( \mathbf{r} = \mathbf{a} + \lambda \mathbf{b} \) uses \( \mathbf{a} \) as a point on the line and \( \mathbf{b} \) as a direction vector.
• Vectors underscore a line's path through space, defining its orientation and specific trajectory based on the scalar parameter \( \lambda \).

In our example, each line's direction is influenced by a cross product, showing how these vector lines will extend infinitely unless altered by another vector or condition like intersection.

Orthogonality

Orthogonality, in vector terms, means two vectors are at right angles to each other. For vectors \( \mathbf{u} \) and \( \mathbf{v} \), this means \( \mathbf{u} \cdot \mathbf{v} = 0 \), where \( \cdot \) denotes the dot product. This concept is the foundation for the cross product as the resultant vector is orthogonal to the initial pair.

• Vectors that are orthogonal have no influence in each other's direction.
• In terms of cross product, the resultant orthogonal vector leads to information about the plane formed by initial vectors.

Orthogonality is crucial in understanding perpendicular relationships in vector geometry, such as when considering the normal to a surface being perpendicular to tangent vectors at all points on the surface.