Introduction to Three-dimensional Geometry revision notes

Key Concepts & Definitions

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.

XY-plane:

The plane containing the x and y axes.

YZ-plane:

The plane containing the y and z axes.

ZX-plane:

The plane containing the z and x axes.

JEE TIPThe perpendicular distance of any point P(x,y,z)P(x, y, z)P(x,y,z) from the YZ, ZX, and XY planes is given by ∣x∣,∣y∣|x|, |y|∣x∣,∣y∣, and ∣z∣|z|∣z∣ respectively.

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.

• Octant I (XOYZ): (+, +, +)
• Octant II (X'OYZ): (-, +, +)
• Octant III (X'OY'Z): (-, -, +)
• Octant IV (XOY'Z): (+, -, +)
• Octant V (XOYZ'): (+, +, -)
• Octant VI (X'OYZ'): (-, +, -)
• Octant VII (X'OY'Z'): (-, -, -)
• Octant VIII (XOY'Z'): (+, -, -)

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

• Direction Cosines ($l, m, n$): The cosines of the angles ($\alpha, \beta, \gamma$) made by a directed line with the positive directions of the x, y, and z-axes respectively.
• Direction Ratios ($a, b, c$): Any three numbers that are proportional to the direction cosines of a line.
• JEE TIPTwo parallel lines have the exact same direction cosines, but their direction ratios can be any proportional scalar multiples.

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.

Lines in 3D Space

1. Equation of a Line

• Point-Parallel Form: The equation of a line passing through a point with position vector $\vec{a}$ and parallel to a vector $\vec{b}$ is $\vec{r} = \vec{a} + \lambda\vec{b}$.
  • Cartesian: $\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c} = \lambda$
• Two-Point Form: The equation of a line passing through two points $\vec{a}$ and $\vec{b}$ is $\vec{r} = \vec{a} + \lambda(\vec{b} - \vec{a})$.
  • Cartesian: $\frac{x-x_1}{x_2-x_1} = \frac{y-y_1}{y_2-y_1} = \frac{z-z_1}{z_2-z_1}$

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 TIPFor 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 $(\vec{b_1} \times \vec{b_2})$.

Planes in 3D Space

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.
• Point-Normal Form: A plane passing through point $\vec{a}$ and perpendicular to vector $\vec{n}$ is $(\vec{r} - \vec{a}) \cdot \vec{n} = 0$.
  • Cartesian: $a(x-x_1) + b(y-y_1) + c(z-z_1) = 0$
• 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 TIPAlways 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 TIPThe 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).

Conditions & Limitations

• 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 TIPIn 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: $SD = \frac{|(\vec{a_2} - \vec{a_1}) \times \vec{b}|}{|\vec{b}|}$.

Previous Year JEE Topics

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 $= 0$).

Memory Aids & JEE Traps

• JEE TIPWhen finding the angle between a line and a plane, remember that the dot product formula $\frac{\vec{b} \cdot \vec{n}}{|\vec{b}||\vec{n}|}$ yields $\sin(\theta)$, NOT $\cos(\theta)$. This is because $\vec{n}$ is the normal to the plane, so the angle between the line and the normal is $90^\circ - \theta$.
• JEE TIPTo find the perpendicular distance from the origin to a plane, the equation $ax+by+cz=d$ MUST be normalized. The distance is $\frac{d}{\sqrt{a^2+b^2+c^2}}$, not just $d$.
• JEE TIPWhen extracting direction ratios from a line equation, ensure the coefficients of $x, y, z$ are strictly $+1$. Example: If a line is $2x - 1 = 3y + 2 = z$, you must rewrite it as $\frac{x - 1/2}{1/2} = \frac{y + 2/3}{1/3} = \frac{z}{1}$ before reading the direction ratios $(1/2, 1/3, 1)$.

Top 10 MCQ Traps

• [JEE TIP] Trap 1 - The Axis Distance Illusion:

  • Misconception: The shortest distance of a spatial point $(x,y,z)$ from the x-axis is simply its $x$-coordinate.
  • Correct Understanding: The coordinate $x$ represents the perpendicular distance of the point from the YZ-plane. The true shortest distance of the point from the x-axis depends entirely on the remaining coordinates and is given by $\sqrt{y^2 + z^2}$.

• [JEE TIP] Trap 2 - The Line-Plane Angle Trigonometric Switch:

  • Misconception: The angle $\theta$ between a straight line with direction vector $\vec{b}$ and a flat plane with normal vector $\vec{n}$ is computed using the standard cosine dot-product formula $\cos\theta = \frac{\vec{b} \cdot \vec{n}}{|\vec{b}||\vec{n}|}$.
  • Correct Understanding: The standard dot product calculates the angle between the line and the normal vector of the plane. Because the true angle of interest is between the line and the surface of the plane, it is complementary ($90^\circ - \theta$). Therefore, the calculation must strictly use $\sin\theta = \frac{\vec{b} \cdot \vec{n}}{|\vec{b}||\vec{n}|}$.

• [JEE TIP] Trap 3 - Cosines vs. Ratios Identity Lock:

  • Misconception: For any arbitrary triplet of numbers $(a, b, c)$ representing the directional components of a vector, the sum of their squares satisfies the identity $a^2 + b^2 + c^2 = 1$.
  • Correct Understanding: This mathematical identity holds strictly for Direction Cosines ($l, m, n$), where $l^2 + m^2 + n^2 = 1$. Ordinary Direction Ratios ($a, b, c$) are scalar multiples of cosines and do not satisfy this condition unless the vector is a unit vector.

• [JEE TIP] Trap 4 - The Collinearity Efficiency Bottleneck:

  • Misconception: Utilizing the 3D distance formula to verify if $AB + BC = AC$ is the most reliable and efficient way to check the collinearity of three points.
  • Correct Understanding: The distance formula involves nested square roots, which are highly time-consuming and prone to calculation errors under exam pressure. A much faster method is to check if the Direction Ratios of vector $\vec{AB}$ and vector $\vec{BC}$ are proportional, or to verify if their vector cross product is zero ($\vec{AB} \times \vec{BC} = \vec{0}$).

• [JEE TIP] Trap 5 - Skew vs. Parallel Formula Blindness:

  • Misconception: The standard shortest distance formula for skew lines ($\frac{|(\vec{a_2}-\vec{a_1}) \cdot (\vec{b_1} \times \vec{b_2})|}{|\vec{b_1} \times \vec{b_2}|}$) can be applied blindly to find the distance between any two non-intersecting lines in space.
  • Correct Understanding: If the two lines are parallel, their direction vectors are identical ($\vec{b_1} = \vec{b_2}$), which makes the cross product $\vec{b_1} \times \vec{b_2} = \vec{0}$ and reduces the denominator to zero. You must always check if $\vec{b_1} \times \vec{b_2} \neq \vec{0}$ first. If they are parallel, switch to the parallel distance formula: $\frac{|(\vec{a_2}-\vec{a_1}) \times \vec{b}|}{|\vec{b}|}$.

• [JEE TIP] Trap 6 - The Hidden Negative Direction Ratio:

  • Misconception: In a symmetrical line equation written as $\frac{3-x}{2} = \frac{y+1}{4}$, the direction ratio corresponding to the $x$-component is directly read as $2$.
  • Correct Understanding: To correctly extract direction ratios, the line equation must strictly match the standard form where the variable coefficients are $+1$, i.e., $(x - x_1)$. Factoring out a negative from the numerator transforms the term into $\frac{x-3}{-2}$. Thus, the actual direction ratio for the $x$-component is $-2$, not $2$.

• [JEE TIP] Trap 7 - Diagrammatic Perpendicular Boundaries:

  • Misconception: The coordinates of the foot of a perpendicular dropped from an external point $(x_1, y_1, z_1)$ onto a plane must always fall inside the finite geometric boundaries sketched on rough paper.
  • Correct Understanding: Geometrically and algebraically, planes extend infinitely in all directions. The foot of the perpendicular is a mathematical projection that can easily lie far outside the arbitrary "triangle" or "parallelogram" boundaries drawn in a quick visualization sketch. Rely strictly on the algebraic formula.

• [JEE TIP] Trap 8 - The False Origin Intercept:

  • Misconception: A plane equation presented in the form $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 0$ is written in standard intercept form, with intercepts $a$, $b$, and $c$.
  • Correct Understanding: The standard intercept form requires the Right-Hand Side (RHS) of the equation to be strictly equal to $1$ ($\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$). If the RHS is $0$, the plane passes directly through the coordinate origin $(0,0,0)$, making individual non-zero axis intercepts geometrically undefined.

• [JEE TIP] Trap 9 - The 2D vs. 3D Locus Dimension Shift:

  • Misconception: The spatial locus of a point that maintains an equal distance from two fixed points in space is a straight line.
  • Correct Understanding: While this locus forms a straight perpendicular bisector line in a 2D plane, moving into 3D space adds an entire dimension of freedom. In three dimensions, the locus of a point equidistant from two fixed coordinates is an infinite perpendicular bisector plane.

• [JEE TIP] Trap 10 - The Scalar vs. Vector Projection Mix-Up:

  • Misconception: The projection of a vector $\vec{a}$ onto another vector $\vec{b}$ is universally represented as a single scalar number.
  • Correct Understanding: JEE multiple-choice questions frequently exploit the distinction between two different terms. The scalar projection is indeed a magnitude ($\vec{a} \cdot \hat{b}$), but the vector projection retains directional attributes and is expressed as a vector: $(\vec{a} \cdot \hat{b})\hat{b}$. Always check the wording of the question carefully.