Problem 10 - A Complete Resource Book in Mathematics for JEE Main

Question

A straight line \(\mathbf{r}=\mathbf{a}+\lambda \mathbf{b}\) meets the \(p\) lane \(\mathbf{r} \cdot \mathbf{n}=0\) in \(P\). The position vector of \(P\) is (A) \(a+\frac{a \cdot n}{b \cdot n} b\) (B) \(\mathrm{a}-\frac{\mathbf{a} \cdot \mathbf{n}}{\mathbf{b} \cdot \mathbf{n}} \mathbf{b}\) (C) \(\mathrm{a}-\frac{\mathrm{a} \cdot \mathbf{n}}{\mathrm{b} \cdot \mathrm{n}} \mathbf{b}\) (D) none of these

Step-by-Step Solution

Answer

The position vector of \( P \) is given by option (C).

Step 1: Understand the equation of line and plane

The line is given by the vector equation \( \mathbf{r} = \mathbf{a} + \lambda \mathbf{b} \), where \( \mathbf{a} \) is a point on the line, \( \mathbf{b} \) is the direction vector, and \( \lambda \) is a scalar parameter. The plane is given by the equation \( \mathbf{r} \cdot \mathbf{n} = 0 \), where \( \mathbf{n} \) is the normal vector to the plane.

Step 2: Substitute the line equation into the plane equation

Substitute \( \mathbf{r} = \mathbf{a} + \lambda \mathbf{b} \) into the plane equation \( \mathbf{r} \cdot \mathbf{n} = 0 \). This results in \( (\mathbf{a} + \lambda \mathbf{b}) \cdot \mathbf{n} = 0 \), which simplifies to \( \mathbf{a} \cdot \mathbf{n} + \lambda (\mathbf{b} \cdot \mathbf{n}) = 0 \).

Step 3: Solve for \( \lambda \)

Rearrange the equation \( \mathbf{a} \cdot \mathbf{n} + \lambda (\mathbf{b} \cdot \mathbf{n}) = 0 \) to find \( \lambda \). We get \( \lambda = -\frac{\mathbf{a} \cdot \mathbf{n}}{\mathbf{b} \cdot \mathbf{n}} \). This is the specific value of \( \lambda \) at which the line intersects the plane.

Step 4: Find the position vector of the point of intersection \( P \)

Substitute \( \lambda = -\frac{\mathbf{a} \cdot \mathbf{n}}{\mathbf{b} \cdot \mathbf{n}} \) back into the line equation \( \mathbf{r} = \mathbf{a} + \lambda \mathbf{b} \) to find the position vector of \( P \). This gives us \( \mathbf{r} = \mathbf{a} - \frac{\mathbf{a} \cdot \mathbf{n}}{\mathbf{b} \cdot \mathbf{n}} \mathbf{b} \).

Step 5: Compare with the given options

The resulting position vector \( \mathbf{r} = \mathbf{a} - \frac{\mathbf{a} \cdot \mathbf{n}}{\mathbf{b} \cdot \mathbf{n}} \mathbf{b} \) matches option (C). Therefore, this is the position vector of point \( P \) where the line meets the plane.

Key Concepts

Equation of a Line

The equation of a line in vector algebra is a fundamental concept. It is often represented as \( \mathbf{r} = \mathbf{a} + \lambda \mathbf{b} \), where \( \mathbf{r} \) is the position vector of any point on the line.
- \( \mathbf{a} \) is a fixed point on the line. This is known as the position vector for a particular point.- \( \mathbf{b} \) is the direction vector, indicating the direction and orientation of the line.- \( \lambda \) is a scalar parameter that can vary to generate every point along the line.
This form allows us to express any line in terms of vector addition and scaling. By changing the value of \( \lambda \), you can find all possible points along the line without changing its direction.

Equation of a Plane

The equation of a plane in vector form is expressed as \( \mathbf{r} \cdot \mathbf{n} = d \). This description uses vector dot product and conveys spatial information about the plane.
- Here, \( \mathbf{r} \) represents the position vector of any point on the plane.- The vector \( \mathbf{n} \) is known as the normal vector, which is perpendicular to every point on the plane.- The term \( d \) in our exercise is set to zero, implying the plane passes through the origin.
A crucial aspect of this equation is its reliance on the dot product, which measures cosines of angles between vectors, effectively capturing how 'aligned' they are in space.

Intersection of Line and Plane

Determining the intersection of a line and a plane involves substituting the line's equation into the plane's equation. It involves a few steps to find the exact point where they meet.
1. Substitute \( \mathbf{r} = \mathbf{a} + \lambda \mathbf{b} \) from the line equation into the plane equation \( \mathbf{r} \cdot \mathbf{n} = 0 \).2. This leads to \( (\mathbf{a} + \lambda \mathbf{b}) \cdot \mathbf{n} = 0 \), simplifying to \( \mathbf{a} \cdot \mathbf{n} + \lambda (\mathbf{b} \cdot \mathbf{n}) = 0 \).3. Solve for \( \lambda \) to get the parameter at which the line intersects the plane: \( \lambda = -\frac{\mathbf{a} \cdot \mathbf{n}}{\mathbf{b} \cdot \mathbf{n}} \).
4. Substitute this value of \( \lambda \) back in the line equation to find the exact position vector at the intersection point \( P \).
By following these steps, one can determine the precise point where the line pierces through the plane.

Vector Dot Product

The vector dot product is a key operation in vector algebra. It is used to find various geometric properties, such as the angle between two vectors or the intersection of geometric shapes.
- The dot product of two vectors \( \mathbf{a} \) and \( \mathbf{b} \) is denoted as \( \mathbf{a} \cdot \mathbf{b} \) and is calculated as \( a_1b_1 + a_2b_2 + a_3b_3 \) for three-dimensional vectors.- It results in a scalar, reflecting the magnitude of projection of one vector onto another.
In the context of planes and lines, the dot product allows us to understand how a direction vector (from a line) aligns with the normal vector of a plane. This is vital for determining intersection points or understanding relationships between different vectors in space. Through this mechanism, vector algebra is fundamentally linked with geometric interpretations.