Problem 33 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The equation of the plane containing the lines \(\mathbf{r}=\mathbf{a}_{1}+\) \(\lambda \mathbf{b}\) and \(\mathbf{r}=\mathbf{a}_{2}+\mu \mathbf{b}\) is (A) \(\mathbf{r} \cdot\left(\mathbf{a}_{1}-\mathbf{a}_{2}\right) \times \mathbf{b}=\left[\mathbf{a}_{1} \mathbf{a}_{2} \mathbf{b}\right]\) (B) \(\mathbf{r} \cdot\left(\mathbf{a}_{2}-\mathbf{a}_{1}\right) \times \mathbf{b}=\left[\mathbf{a}_{1} \mathbf{a}_{2} \mathbf{b}\right]\) (C) \(\mathbf{r} \cdot\left(\mathbf{a}_{1}+\mathbf{a}_{2}\right) \times \mathbf{b}=\left[\mathbf{a}_{2} \mathbf{a}_{1} \mathbf{b}\right]\) (D) none of these

Step-by-Step Solution

Answer

Option (B) is correct: \( \mathbf{r} \cdot (\mathbf{a}_2 - \mathbf{a}_1) \times \mathbf{b} = \left[\mathbf{a}_1 \mathbf{a}_2 \mathbf{b}\right] \).

Step 1: Understand the Plane's Definition

A plane containing two lines must include both their direction vectors and the relative position vector between any points on the lines. If the lines share the same direction vector, this scenario applies to them.

Step 2: Identify the Direction Vector

The direction vector common to both lines is denoted as \( \mathbf{b} \). This vector is included in the plane's definition, as both lines are parallel and lie on the plane.

Step 3: Determine the Point on the Plane

A specific point on the plane can be any point from either line. Select \( \mathbf{a}_1 \) from the first line \( \mathbf{r} = \mathbf{a}_1 + \lambda \mathbf{b} \).

Step 4: Calculate the Normal Vector

The normal vector to the plane is given by the cross product of two vectors lying on the plane. Here, the vectors are \( (\mathbf{a}_2 - \mathbf{a}_1) \) and \( \mathbf{b} \). Hence, the normal vector is \( (\mathbf{a}_2 - \mathbf{a}_1) \times \mathbf{b} \).

Step 5: Formulate the Plane Equation

The equation of a plane can be written as \( \mathbf{r} \cdot \mathbf{n} = \mathbf{a} \cdot \mathbf{n} \), where \( \mathbf{n} \) is the normal vector and \( \mathbf{a} \) is a point on the plane. Thus, the equation is \( \mathbf{r} \cdot ((\mathbf{a}_2 - \mathbf{a}_1) \times \mathbf{b}) = \mathbf{a}_1 \cdot ((\mathbf{a}_2 - \mathbf{a}_1) \times \mathbf{b}) \).

Step 6: Simplify Using Scalar Triple Product

The right side of the equation can be simplified using the scalar triple product, resulting in \( [\mathbf{a}_1 \mathbf{a}_2 \mathbf{b}] \). Therefore, the plane equation becomes \( \mathbf{r} \cdot ((\mathbf{a}_2 - \mathbf{a}_1) \times \mathbf{b}) = [\mathbf{a}_1 \mathbf{a}_2 \mathbf{b}] \).

Step 7: Choose the Correct Option

Comparing the derived equation with the given options, option (B) matches: \( \mathbf{r} \cdot (\mathbf{a}_2 - \mathbf{a}_1) \times \mathbf{b} = \left[\mathbf{a}_1 \mathbf{a}_2 \mathbf{b}\right] \).

Key Concepts

Vector Equations

In 3D geometry, vector equations describe lines or planes using vectors. For a line, a vector equation is typically given as \( \mathbf{r} = \mathbf{a} + \lambda \mathbf{b} \), where \( \mathbf{r} \) represents any point on the line. Here, \( \mathbf{a} \) is a fixed point on the line, and \( \mathbf{b} \) is the direction vector of the line. \( \lambda \) is a scalar that can vary to give all points along the line.
Key aspects of vector equations:

• Provides a way to represent lines and planes in a flexible, algebraic form.
• Helps visualize geometric data in a multi-dimensional space.

To solve problems involving vector equations, one often needs to manipulate these vectors and scalars to find intersections, angles, or constructs like planes. Understanding these linear algebra components is crucial for tackling 3D geometry questions.

Plane Equations

A plane equation, much like a line equation, serves the purpose of establishing a mathematical representation for any plane in three-dimensional space. A common form of a plane equation uses a point \( \mathbf{a} \) on the plane and a normal vector \( \mathbf{n} \) perpendicular to the plane. It is of the form:
\[ \mathbf{r} \cdot \mathbf{n} = \mathbf{a} \cdot \mathbf{n} \]
This compact form helps determine if a point lies on the plane by checking if it satisfies the equation.
Steps to formulating a plane equation:

• Determine a point on the plane, often derived from a vector equation.
• Find the normal vector, especially through cross products of vectors lying on the plane.

A practical aspect is using vector arithmetic, such as dot and cross products, to determine orientational relationships, which are integral to forming the equation.

Scalar Triple Product

The scalar triple product is a fundamental tool used within the study of vectors and planes, especially in 3D geometry. The scalar triple product of vectors \( \mathbf{a}, \mathbf{b}, \mathbf{c} \), represented as \( [\mathbf{a} \mathbf{b} \mathbf{c}] \), is calculated as:
\[ [\mathbf{a} \mathbf{b} \mathbf{c}] = \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \]
This calculation yields a scalar value and is deeply linked to the geometric properties of the vectors involved.
Applications include:

• Determining the volume of the parallelepiped formed by the three vectors.
• Checking if the vectors are coplanar; if the scalar result is zero, the vectors lie in the same plane.

In practical use, this product offers a robust way to interpret interactions among three vectors, such as those forming the normal vector in plane equations. Understanding how to perform and apply the scalar triple product simplifies many geometric problem-solving processes in 3D space.