Three Dimensional Geometry

• Axes and Octants: The three mutually perpendicular lines passing through the origin $O$ are the $X$, $Y$, and $Z$ axes. These axes divide the 3D space into $8$ regions called octants. The signs of the coordinates $(x, y, z)$ determine the octant:

  Octant	I	II	III	IV	V	VI	VII	VIII
  $x$	$+$	$-$	$-$	$+$	$+$	$-$	$-$	$+$
  $y$	$+$	$+$	$-$	$-$	$+$	$+$	$-$	$-$
  $z$	$+$	$+$	$+$	$+$	$-$	$-$	$-$	$-$

• Distance Formula: The distance $d$ between two points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ is given by: $$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$$

• Section Formula: If a point $R$ divides the line segment joining $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ in the ratio $m:n$, its coordinates are:
  • Internal Division: $$R \equiv \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n} \right)$$
  • External Division: $$R \equiv \left( \frac{mx_2 - nx_1}{m-n}, \frac{my_2 - ny_1}{m-n}, \frac{mz_2 - nz_1}{m-n} \right)$$
• Centroid of a Triangle: For vertices $A(x_1,y_1,z_1)$, $B(x_2,y_2,z_2)$, and $C(x_3,y_3,z_3)$: $$G \equiv \left( \frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}, \frac{z_1+z_2+z_3}{3} \right)$$
• Centroid of a Tetrahedron: For vertices $A_i(x_i, y_i, z_i)$ where $i = 1, 2, 3, 4$: $$G \equiv \left( \frac{\sum x_i}{4}, \frac{\sum y_i}{4}, \frac{\sum z_i}{4} \right)$$

• Direction Cosines: If a line makes angles $\alpha, \beta, \gamma$ with the positive direction of the $X, Y, Z$ axes respectively, then $l = \cos\alpha$, $m = \cos\beta$, $n = \cos\gamma$ are the direction cosines.
• Fundamental Identity: $$l^2 + m^2 + n^2 = 1 \implies \cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1$$
• Direction Ratios: Any three numbers $a, b, c$ proportional to the direction cosines $l, m, n$ are called direction ratios. Thus, $\frac{l}{a} = \frac{m}{b} = \frac{n}{c} = k$.
• Conversion from DRs to DCs: $$l = \frac{\pm a}{\sqrt{a^2+b^2+c^2}}, \quad m = \frac{\pm b}{\sqrt{a^2+b^2+c^2}}, \quad n = \frac{\pm c}{\sqrt{a^2+b^2+c^2}}$$

• Angle ($\theta$) between lines: $$\cos\theta = |l_1l_2 + m_1m_2 + n_1n_2| = \frac{|a_1a_2 + b_1b_2 + c_1c_2|}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}$$ $$\sin\theta = \sqrt{\sum (l_1m_2 - l_2m_1)^2}$$
• Condition for Perpendicular Lines: $\theta = 90^\circ$ $$l_1l_2 + m_1m_2 + n_1n_2 = 0 \quad \text{or} \quad a_1a_2 + b_1b_2 + c_1c_2 = 0$$
• Condition for Parallel Lines: $\theta = 0^\circ$ $$\frac{l_1}{l_2} = \frac{m_1}{m_2} = \frac{n_1}{n_2} \quad \text{or} \quad \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$$

• Symmetric/Cartesian Form: Passing through $(x_0, y_0, z_0)$ with DRs $(a, b, c)$: $$\frac{x-x_0}{a} = \frac{y-y_0}{b} = \frac{z-z_0}{c}$$
• Vector Form: Passing through position vector $\mathbf{a}$ and parallel to vector $\mathbf{b} = a\hat{\mathbf{i}} + b\hat{\mathbf{j}} + c\hat{\mathbf{k}}$: $$\mathbf{r} = \mathbf{a} + \lambda\mathbf{b}$$
• Parametric Form: Setting the Cartesian form equal to a parameter $\lambda$: $$x = x_0 + a\lambda, \quad y = y_0 + b\lambda, \quad z = z_0 + c\lambda$$ Any general point on the line can be represented as $(x_0 + a\lambda, y_0 + b\lambda, z_0 + c\lambda)$.
• Intersection of Two Lines: To find if lines $\mathbf{r} = \mathbf{a}_1 + \lambda\mathbf{b}_1$ and $\mathbf{r} = \mathbf{a}_2 + \mu\mathbf{b}_2$ intersect, equate their parametric forms. If a unique pair of values for $\lambda$ and $\mu$ satisfies all three coordinate equations, the lines intersect.

• Skew Lines: Lines that are neither parallel nor intersecting and lie in different planes.
• Shortest Distance (SD) for Skew Lines: Between $\mathbf{r} = \mathbf{a}_1 + \lambda\mathbf{b}_1$ and $\mathbf{r} = \mathbf{a}_2 + \mu\mathbf{b}_2$: $$d = \frac{|(\mathbf{a}_2 - \mathbf{a}_1) \cdot (\mathbf{b}_1 \times \mathbf{b}_2)|}{|\mathbf{b}_1 \times \mathbf{b}_2|}$$ In Cartesian form, the numerator is evaluated as the absolute value of the determinant: $$\left| \begin{matrix} x_2-x_1 & y_2-y_1 & z_2-z_1 \\ a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \end{matrix} \right|$$
• Condition for Intersecting/Coplanar Lines: The shortest distance is zero: $$(\mathbf{a}_2 - \mathbf{a}_1) \cdot (\mathbf{b}_1 \times \mathbf{b}_2) = 0$$
• Distance Between Parallel Lines: Between $\mathbf{r} = \mathbf{a}_1 + \lambda\mathbf{b}$ and $\mathbf{r} = \mathbf{a}_2 + \mu\mathbf{b}$: $$d = \frac{|(\mathbf{a}_2 - \mathbf{a}_1) \times \mathbf{b}|}{|\mathbf{b}|}$$

• General Form: $ax + by + cz + d = 0$, where $(a, b, c)$ are the DRs of the normal vector to the plane.
• Vector Form: Normal to $\mathbf{n}$ and passing through $\mathbf{a}$: $$(\mathbf{r} - \mathbf{a}) \cdot \mathbf{n} = 0 \implies \mathbf{r} \cdot \mathbf{n} = \mathbf{a} \cdot \mathbf{n}$$
• Normal Form: $\mathbf{r} \cdot \hat{\mathbf{n}} = p$ or $lx + my + nz = p$, where $\hat{\mathbf{n}}$ is the unit normal vector and $p$ is the perpendicular distance from the origin ($p \ge 0$).
• Intercept Form: Making intercepts $a, b, c$ on the $X, Y, Z$ axes respectively: $$\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$$
• Plane through Three Non-Collinear Points: $P(x_1, y_1, z_1)$, $Q(x_2, y_2, z_2)$, and $R(x_3, y_3, z_3)$: $$\left| \begin{matrix} x-x_1 & y-y_1 & z-z_1 \\ x_2-x_1 & y_2-y_1 & z_2-z_1 \\ x_3-x_1 & y_3-y_1 & z_3-z_1 \end{matrix} \right| = 0$$

• Angle ($\theta$) between two planes: Given $a_1x + b_1y + c_1z + d_1 = 0$ and $a_2x + b_2y + c_2z + d_2 = 0$: $$\cos\theta = \frac{|a_1a_2 + b_1b_2 + c_1c_2|}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}$$
• Perpendicular Planes: $a_1a_2 + b_1b_2 + c_1c_2 = 0$
• Parallel Planes: $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$

• Distance of a Point from a Plane: Distance of $P(x_1, y_1, z_1)$ from $ax + by + cz + d = 0$: $$D = \frac{|ax_1 + by_1 + cz_1 + d|}{\sqrt{a^2+b^2+c^2}}$$
• Foot of Perpendicular ($H$) from a point to a plane: $$\frac{x_H - x_1}{a} = \frac{y_H - y_1}{b} = \frac{z_H - z_1}{c} = -\frac{(ax_1 + by_1 + cz_1 + d)}{a^2+b^2+c^2}$$
• Image of a Point ($I$) in a plane: $$\frac{x_I - x_1}{a} = \frac{y_I - y_1}{b} = \frac{z_I - z_1}{c} = -\frac{2(ax_1 + by_1 + cz_1 + d)}{a^2+b^2+c^2}$$
• Angle between a Line and a Plane: Angle $\phi$ between line $\mathbf{r} = \mathbf{a} + \lambda\mathbf{b}$ and plane $\mathbf{r} \cdot \mathbf{n} = d$: $$\sin\phi = \frac{|\mathbf{b} \cdot \mathbf{n}|}{|\mathbf{b}||\mathbf{n}|}$$
  • Line parallel to plane: $\mathbf{b} \cdot \mathbf{n} = 0 \implies aa_1 + bb_1 + cc_1 = 0$
  • Line perpendicular to plane: $\mathbf{b} \times \mathbf{n} = \mathbf{0} \implies \frac{a}{a_1} = \frac{b}{b_1} = \frac{c}{c_1}$
  • Line lying entirely in the plane: $\mathbf{b} \cdot \mathbf{n} = 0$ AND the passing point $\mathbf{a}$ satisfies the plane equation.
• Coplanarity of Two Lines: Lines $\frac{x-x_1}{a_1} = \frac{y-y_1}{b_1} = \frac{z-z_1}{c_1}$ and $\frac{x-x_2}{a_2} = \frac{y-y_2}{b_2} = \frac{z-z_2}{c_2}$ are coplanar if: $$\left| \begin{matrix} x_2-x_1 & y_2-y_1 & z_2-z_1 \\ a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \end{matrix} \right| = 0$$ The equation of the plane containing them is: $$\left| \begin{matrix} x-x_1 & y-y_1 & z-z_1 \\ a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \end{matrix} \right| = 0$$
• Family of Planes: Equation of a plane passing through the line of intersection of planes $P_1 = 0$ and $P_2 = 0$ is: $$P_1 + \lambda P_2 = 0 \implies (a_1x+b_1y+c_1z+d_1) + \lambda(a_2x+b_2y+c_2z+d_2) = 0$$

• General Equation: $x^2 + y^2 + z^2 + 2ux + 2vy + 2wz + d = 0$
  • Centre: $(-u, -v, -w)$
  • Radius: $R = \sqrt{u^2 + v^2 + w^2 - d}$ (Condition for real sphere: $u^2+v^2+w^2 \ge d$)
• Diameter Form: Sphere with diametric endpoints $A(x_1,y_1,z_1)$ and $B(x_2,y_2,z_2)$: $$(x-x_1)(x-x_2) + (y-y_1)(y-y_2) + (z-z_1)(z-z_2) = 0$$
• Intersection with a Plane: The intersection of a sphere and a plane is a circle.
  • Let $C$ be the center of the sphere, $R$ be its radius, and $p$ be the perpendicular distance from $C$ to the intersecting plane.
  • Radius of resulting circle: $r = \sqrt{R^2 - p^2}$
  • Tangent Plane condition: If $p = R$, the plane touches the sphere at a single point. The equation of the tangent plane at point $(x_1, y_1, z_1)$ on the sphere is: $$xx_1 + yy_1 + zz_1 + u(x+x_1) + v(y+y_1) + w(z+z_1) + d = 0$$

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