3D Geometry Notes

Introduction

3D Geometry is a crucial topic in the JEE Main Mathematics syllabus. It involves understanding and visualizing objects in three dimensions, which is essential for solving problems related to lines, planes, and their intersections. This study note will break down the complexities of 3D Geometry into manageable sections, ensuring a clear understanding of each concept.

Coordinate System in 3D

The 3D coordinate system extends the 2D Cartesian system by adding a third axis, the z-axis, perpendicular to both the x-axis and y-axis.

Points in 3D Space

A point in 3D space is represented as $(x, y, z)$, where $x$, $y$, and $z$ are the coordinates along the x-axis, y-axis, and z-axis, respectively.

Distance Formula

The distance between two points $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$ is given by:

$$ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} $$

Section Formula

The coordinates of a point dividing the line segment joining $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$ in the ratio $m:n$ are:

$$ \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n} \right) $$

Direction Ratios and Direction Cosines

Direction Ratios

Direction ratios (DRs) of a line are proportional to the differences in the coordinates of any two points on the line.

Direction Cosines

Direction cosines (DCs) are the cosines of the angles that a line makes with the coordinate axes. If $\alpha$, $\beta$, and $\gamma$ are the angles with the x-axis, y-axis, and z-axis, respectively, then the direction cosines are $\cos \alpha$, $\cos \beta$, and $\cos \gamma$.

The relationship between direction cosines is:

$$ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 $$

Equations of Lines in 3D

There are various forms to represent the equation of a line in 3D space.

Vector Form

If $\mathbf{a}$ is a position vector of a point on the line and $\mathbf{b}$ is a direction vector, then the vector equation of the line is:

$$ \mathbf{r} = \mathbf{a} + \lambda \mathbf{b} $$

Cartesian Form

If a line passes through a point $(x_1, y_1, z_1)$ and has direction ratios $a, b, c$, its Cartesian equation is:

$$ \frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c} $$

Equations of Planes in 3D

General Form

The general equation of a plane is:

$$ ax + by + cz + d = 0 $$

Normal Form

If the normal to the plane makes angles $\alpha$, $\beta$, and $\gamma$ with the x, y, and z axes, and the perpendicular distance from the origin to the plane is $p$, then the equation is:

$$ x \cos \alpha + y \cos \beta + z \cos \gamma = p $$

Intercept Form

If a plane intercepts the x, y, and z axes at $a$, $b$, and $c$ respectively, its equation is:

$$ \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 $$

Angle Between Two Lines

If two lines have direction cosines $(l_1, m_1, n_1)$ and $(l_2, m_2, n_2)$, the cosine of the angle $\theta$ between them is given by:

$$ \cos \theta = l_1 l_2 + m_1 m_2 + n_1 n_2 $$

Angle Between a Line and a Plane

If a line with direction ratios $a, b, c$ makes an angle $\theta$ with a plane with normal vector $\mathbf{n} = (A, B, C)$, then:

$$ \sin \theta = \frac{|aA + bB + cC|}{\sqrt{A^2 + B^2 + C^2} \sqrt{a^2 + b^2 + c^2}} $$

Distance of a Point from a Plane

The perpendicular distance of a point $(x_1, y_1, z_1)$ from the plane $ax + by + cz + d = 0$ is:

$$ \frac{|ax_1 + by_1 + cz_1 + d|}{\sqrt{a^2 + b^2 + c^2}} $$

Conclusion

Understanding 3D Geometry is essential for solving complex problems in JEE Main Mathematics. By mastering the concepts of points, lines, and planes in three dimensions, students can approach these problems with confidence and precision. Practice regularly and make use of these study notes to reinforce your understanding.