Problem 12 - A Complete Resource Book in Mathematics for JEE Main

Question

The length of the perpendicular from the origin to the plane passing though three non-collinear points \(\mathbf{a}, \mathbf{b}, \mathbf{c}\) is (A) \(\frac{[\mathrm{abc}]}{|\mathbf{a} \times \mathbf{b}+\mathbf{c} \times \mathbf{a}+\mathbf{b} \times \mathbf{c}|}\) (B) \(\frac{2[\mathbf{a} \mathbf{b c}]}{|\mathbf{a} \times \mathbf{b}+\mathbf{b} \times \mathbf{c}+\mathbf{c} \times \mathbf{a}|}\) (C) \([\mathbf{a} \mathbf{b} \mathbf{c}]\) (D) none of these

Step-by-Step Solution

Answer

The answer is (B).

Step 1: Understand the Problem

We need to find the length of the perpendicular from the origin to a plane defined by three non-collinear points \( \mathbf{a}, \mathbf{b}, \mathbf{c} \). The equation of the plane can be determined using these points.

Step 2: Find the Normal Vector of the Plane

The normal vector \( \mathbf{n} \) of the plane can be found using the cross product of vectors formed by these points, i.e., \( \mathbf{n} = (\mathbf{b} - \mathbf{a}) \times (\mathbf{c} - \mathbf{a}) \). This vector is perpendicular to the plane.

Step 3: Write the Plane Equation

Using the normal vector \( \mathbf{n} \) and point \( \mathbf{a} \), the plane equation is given by \( \mathbf{n} \cdot (\mathbf{r} - \mathbf{a}) = 0 \), where \( \mathbf{r} \) is a general position vector on this plane.

Step 4: Simplify the Determinant Expression

Recall that \( [\mathbf{a} \mathbf{b} \mathbf{c}] \) represents the scalar triple product of vectors \( \mathbf{a}, \mathbf{b}, \mathbf{c} \). It gives the volume of the parallelepiped formed by them. The determinant \( [\mathbf{a} \mathbf{b} \mathbf{c}] = \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \).

Step 5: Calculate the Length of Perpendicular

The formula for the length of the perpendicular from the origin to the plane is \( \frac{d}{|\mathbf{n}|} \), where \( d \) is the perpendicular distance from the origin (0), and \( \mathbf{n} \) is the normal vector. Given \( d = \left| \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \right| \), the formula becomes \( \frac{|[\mathbf{a} \mathbf{b} \mathbf{c}]|}{|\mathbf{a}\times \mathbf{b} + \mathbf{b}\times \mathbf{c} + \mathbf{c}\times \mathbf{a}|} \), which is exactly (B).

Key Concepts

Planes in 3D

In 3D geometry, planes are flat surfaces extending indefinitely in two dimensions. The equation of a plane can typically be defined using a point-position vector on the plane and a normal vector perpendicular to the surface. For any point on a plane

• the equation is given as \( \mathbf{n} \cdot (\mathbf{r} - \mathbf{a}) = 0 \),
• where \( \mathbf{n} \) is the normal vector, \( \mathbf{r} \) is a general position vector, and \( \mathbf{a} \) is any known point on the plane.

This equation translates to saying that the vector from any point on the plane to the known point on the plane is perpendicular to the normal vector. This principle helps in understanding what defines a plane in 3D space, making it easier to solve problems related to distances from points (such as the origin) to planes.Planes can also be defined using three non-collinear points. Given points \( \mathbf{a}, \mathbf{b}, \mathbf{c} \), one can find vectors \( \mathbf{b}-\mathbf{a} \) and \( \mathbf{c}-\mathbf{a} \) that form a parallelogram on the plane. The cross product of these vectors gives a vector normal to the plane.

Cross Product

The cross product is a key concept in vector mathematics, especially in 3D spaces. It provides a way to create a vector that is perpendicular to two given vectors, making it vitally important when finding normal vectors to planes.The cross product of vectors \( \mathbf{u} \) and \( \mathbf{v} \) is defined as:

• \( \mathbf{u} \times \mathbf{v} = \det\begin{pmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \u_1 & u_2 & u_3 \v_1 & v_2 & v_3 \end{pmatrix} \)
• This results in a new vector \( \mathbf{w} \) having components derived from the determinants of the 2x2 matrices.

Cross product is useful not only to determine perpendicular vectors but also to find areas of parallelograms formed by the two vectors. The length of the resulting vector represents the area.

Scalar Triple Product

The scalar triple product is an operation involving three vectors: \( \mathbf{a}, \mathbf{b}, \mathbf{c} \). It evaluates to a scalar which represents the volume of the parallelepiped formed by these vectors. You compute it using the determinant of a 3x3 matrix with the vectors as rows or:

• Calculated as: \( [\mathbf{a} \mathbf{b} \mathbf{c}] = \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \)
• The result is a scalar indicating volume. A zero result signifies that the vectors are coplanar.

This operation is significant in problems involving parallelism and coplanarity. It also helps us assess the perpendicular distance from the origin to a plane defined by vectors, as seen in the solution of the given exercise. Understanding the scalar triple product is crucial for a deeper comprehension of 3D geometry.