Introduction to Three-dimensional Geometry revision notes

1. The Rectangular Coordinate System In three-dimensional space, the position of a point is determined by three mutually perpendicular lines passing through a common point O. These lines are called the coordinate axes: the x-axis (X'OX), the y-axis (Y'OY), and the z-axis (Z'OZ). The point O is called the origin.

2. Coordinate Planes The three axes taken in pairs determine three mutually perpendicular planes called coordinate planes.

3. Octants and Sign Conventions The three coordinate planes divide the entire 3D space into eight distinct regions called octants (denoted as I, II, III, IV, V, VI, VII, VIII). The sign of the coordinates determines the specific octant in which a point lies.

4. Direction Cosines (DC) & Direction Ratios (DR)

5. Historical Context The three coordinate planes used today were introduced by J. Bernoulli in 1715. Antoinne Parent gave a systematic development of analytical solid geometry in 1700, and L. Euler expanded on it systematically in 1748. Rene Descartes had the idea of 3D coordinates but did not develop it.

1. Equation of a Line

2. Angle Between Two Lines

If two lines have direction ratios $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$, the angle $\theta$ between them is given by $\cos\theta = \frac{|a_1a_2 + b_1b_2 + c_1c_2|}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}$.

• JEE Tip For perpendicular lines: $a_1a_2 + b_1b_2 + c_1c_2 = 0$. For parallel lines: $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$.

3. Skew Lines & Shortest Distance

Skew lines are lines in space that are neither parallel nor intersecting. They lie in different planes.

• The shortest distance between lines $\vec{r} = \vec{a_1} + \lambda\vec{b_1}$ and $\vec{r} = \vec{a_2} + \mu\vec{b_2}$ is the projection of $(\vec{a_2} - \vec{a_1})$ along the vector perpendicular to both lines $(\vec{b_1} \times \vec{b_2})$.

1. Equation of a Plane

• Normal Form: $\vec{r} \cdot \hat{n} = d$, where $d$ is the perpendicular distance from the origin and $\hat{n}$ is the unit normal vector.

• Intercept Form: $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$, where $a, b, c$ are the x, y, and z intercepts respectively.

2. Family of Planes

The equation of a plane passing through the line of intersection of two planes $P_1 = 0$ and $P_2 = 0$ is $P_1 + \lambda P_2 = 0$.

• JEE Tip Always evaluate the scalar $\lambda$ by using the additional geometric condition given in the problem (like passing through a specific point or being perpendicular to another plane).

3. Distance and Image of a Point

• The perpendicular distance of a point $(x_1, y_1, z_1)$ from a plane $Ax + By + Cz + D = 0$ is $\frac{|Ax_1 + By_1 + Cz_1 + D|}{\sqrt{A^2 + B^2 + C^2}}$.
• JEE Tip The coordinates of the foot of the perpendicular $(x, y, z)$ from $(x_1, y_1, z_1)$ to the plane is given by the symmetric ratio: $\frac{x-x_1}{A} = \frac{y-y_1}{B} = \frac{z-z_1}{C} = -\frac{Ax_1 + By_1 + Cz_1 + D}{A^2 + B^2 + C^2}$. (To find the image, multiply the right-hand side by 2).

1. Distance Formula The distance $PQ$ between two points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ is $PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$.

• Distance from origin $O(0,0,0)$ to point $Q(x,y,z)$ is $OQ = \sqrt{x^2 + y^2 + z^2}$.

2. Section Formula The coordinates of the point $R$ which divides the line segment joining points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ in the ratio $m:n$ are:

• Internal Division: $\left( \frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n}, \frac{mz_2 + nz_1}{m + n} \right)$
• External Division: $\left( \frac{mx_2 - nx_1}{m - n}, \frac{my_2 - ny_1}{m - n}, \frac{mz_2 - nz_1}{m - n} \right)$

3. Centroid of a Triangle and Tetrahedron

• Triangle: For vertices $(x_1, y_1, z_1)$, $(x_2, y_2, z_2)$, and $(x_3, y_3, z_3)$, the centroid $G$ is $(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}, \frac{z_1+z_2+z_3}{3})$.
• Tetrahedron: For four vertices, the centroid is $(\frac{\sum x_i}{4}, \frac{\sum y_i}{4}, \frac{\sum z_i}{4})$.

4. Direction Cosine Identity For any line with direction cosines $l, m, n$: $l^2 + m^2 + n^2 = 1$ (or $\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1$).

• JEE Tip Note that $\sin^2\alpha + \sin^2\beta + \sin^2\gamma = 2$.

5. Shortest Distance Formula $SD = \frac{|(\vec{a_2} - \vec{a_1}) \cdot (\vec{b_1} \times \vec{b_2})|}{|\vec{b_1} \times \vec{b_2}|}$

Variables: All spatial coordinates ($x, y, z$) and distances are in generalized distance "units". Angles are in radians.

• Distance Formula Check: To prove collinearity using the distance formula, one must show that the sum of distances between two pairs of points equals the third distance (e.g., $PQ + QR = PR$). Limitation: This method is computationally heavy. JEE Tip In JEE, it is much faster to prove collinearity by showing that the direction ratios of $PQ$ and $QR$ are proportional, or that the area of the triangle formed by them (via cross product) is zero.
• Direction Ratios vs Cosines: The identity $a^2 + b^2 + c^2 = 1$ is strictly invalid for Direction Ratios unless they are explicitly normalized into Direction Cosines by dividing by $\sqrt{a^2+b^2+c^2}$.
• Skew Lines Condition: The shortest distance formula for skew lines will yield $0$ if the lines intersect. If the lines are parallel, the cross product $\vec{b_1} \times \vec{b_2} = 0$, making the denominator zero. For parallel lines, use the specific parallel shortest distance formula: $SD = \frac{|(\vec{a_2} - \vec{a_1}) \times \vec{b}|}{|\vec{b}|}$.

• Octant Sign Errors: Assuming that octants follow the 2D quadrant patterns exactly. While Octants I-IV mirror Quadrants I-IV with a positive $z$, Octants V-VIII mirror them with a negative $z$.
• Image through Origin: The image of a point $(x, y, z)$ through the origin is $(-x, -y, -z)$. Its image through the X-axis is $(x, -y, -z)$ – the independent variable keeps its sign, while the others flip.
• Right-Handed System: The vectors $\hat{i}, \hat{j}, \hat{k}$ must strictly follow the right-hand thumb rule where $\hat{i} \times \hat{j} = \hat{k}$. If you arbitrarily assign axes without checking orthogonality and orientation, cross-product derivations will have inverted signs.

Based on the historical frequency in JEE Main and Advanced, the following subtopics are heavily tested:

1. Shortest distance between two skew lines (highest frequency).
2. Image and foot of the perpendicular of a point on a plane or a line.
3. Family of planes intersecting at a line, specifically finding a plane from the family that satisfies a distance condition.
4. Locus of points satisfying a 3D distance relationship (e.g., $PA^2 + PB^2 = 2k^2$).
5. Coplanarity of lines (scalar triple product $= 0$).