Formulas

Basic Vector Operations

• Vector Addition: \[ \vec{u} + \vec{v} = \begin{pmatrix} u_1 + v_1 \\ u_2 + v_2 \end{pmatrix} \]
• Vector Subtraction: \[ \vec{u} - \vec{v} = \begin{pmatrix} u_1 - v_1 \\ u_2 - v_2 \end{pmatrix} \]
• Scalar Multiplication: \[ c\vec{v} = \begin{pmatrix} cv_1 \\ cv_2 \end{pmatrix} \]
• Vector Between Two Points: \(\vec{BA} = A - B\)

Vector Norm and Distance

• Norm of a Vector in \(\mathbb{R}^n\): \[ ||\vec{v}|| = \sqrt{v_1^2 + v_2^2 + \dots + v_n^2} \]
• Distance between Points P and Q: \(d(P, Q) = ||Q - P||\)
• Normalization (Unit Vector): \[ \hat{u} = \frac{\vec{v}}{||\vec{v}||} \]
• Parallelogram Law: \[ ||\vec{a} + \vec{b}||^2 + ||\vec{a} - \vec{b}||^2 = 2(||\vec{a}||^2 + ||\vec{b}||^2) \]
• Magnitude of Vector Difference Squared: \(||\vec{a} - \vec{b}||^2 = ||\vec{a}||^2 - 2(\vec{a}\cdot\vec{b}) + ||\vec{b}||^2\)

Dot Product

• Algebraic Dot Product: \[ \vec{u} \cdot \vec{v} = \sum_{i=1}^{n} u_i v_i \]
• Geometric Dot Product: \[ \vec{v} \cdot \vec{w} = ||\vec{v}|| \cdot ||\vec{w}|| \cos(\theta) \]
• Dot Product with Scalar Multiplication: \(\vec{a} \cdot (k\vec{b}) = k(\vec{a} \cdot \vec{b})\)
• Distributive Property of Dot Product: \(\vec{a} \cdot (\vec{b} + \vec{c}) = \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c}\)
• Dot Product and Norm: \(\vec{v} \cdot \vec{v} = ||\vec{v}||^2\)
• Angle Between Vectors: \[ \cos(\theta) = \frac{\vec{v} \cdot \vec{w}}{||\vec{v}|| \cdot ||\vec{w}||} \]
• Orthogonality Condition: \(\vec{v} \cdot \vec{w} = 0\)

Cross Product and Triple Products

• Cross Product: \[\vec{a} \times \vec{b} = \det\begin{pmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{pmatrix}\]
• Magnitude of Cross Product (Geometric): \(||\vec{u} \times \vec{v}|| = ||\vec{u}|| ||\vec{v}|| \sin(\theta)\)
• Cross Product and Dot Product Identity: \(|\vec{a} \times \vec{b}|^2 + (\vec{a} \cdot \vec{b})^2 = |\vec{a}|^2|\vec{b}|^2\)
• Scalar Triple Product (Volume of Parallelepiped): \(V = |\vec{a} \cdot (\vec{b} \times \vec{c})|\)
• Scalar Triple Product as Determinant: \[\vec{a} \cdot (\vec{b} \times \vec{c}) = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix}\]
• Coplanarity Test: Three vectors are coplanar if \(\vec{a} \cdot (\vec{b} \times \vec{c}) = 0\)
• Vector Triple Product Identity: \(\vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c})\vec{b} - (\vec{a} \cdot \vec{b})\vec{c}\)

Projections and Reflections

• Projection of vector \(\vec{a}\) onto vector \(\vec{b}\): \[ \text{proj}_{\vec{b}}\vec{a} = \frac{\vec{a} \cdot \vec{b}}{||\vec{b}||^2} \vec{b} \]
• Scalar Projection (Component): \[ \text{comp}_{\vec{w}}(\vec{v}) = \frac{\vec{v} \cdot \vec{w}}{||\vec{w}||} \]
• Vector Decomposition: \(\vec{v} = \vec{v}_{||} + \vec{v}_{\perp}\), where \(\vec{v}_{||} = \text{proj}_{\vec{w}}(\vec{v})\) and \(\vec{v}_{\perp} = \vec{v} - \vec{v}_{||}\)
• Reflection of vector \(\vec{a}\) over a line defined by vector \(\vec{b}\): \[ \text{ref}_{\vec{b}}\vec{a} = 2 \cdot \text{proj}_{\vec{b}}\vec{a} - \vec{a} \]

Geometric Formulas

• Midpoint of a Segment: \[ \vec{OM} = \frac{1}{2}(\vec{OB} + \vec{OC}) \]
• Midpoint Formula (2D): \(M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)\)
• Position Vector of Centroid G: \[ \vec{OG} = \frac{1}{3}(\vec{OA} + \vec{OB} + \vec{OC}) \]
• Centroid of a Triangle: \(C = \left(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}\right)\)
• Median Vector: \[ \vec{AD} = \frac{\vec{AB} + \vec{AC}}{2} \]
• Section Formula: \(R = \left(\frac{kx_2+x_1}{k+1}, \frac{ky_2+y_1}{k+1}, \frac{kz_2+z_1}{k+1}\right)\)
• Direction Cosines: \(\cos(\alpha) = \frac{a_1}{||\vec{a}||}, \cos(\beta) = \frac{a_2}{||\vec{a}||}, \cos(\gamma) = \frac{a_3}{||\vec{a}||}\)
• Direction Cosines Identity: \[ \cos^2(\alpha) + \cos^2(\beta) + \cos^2(\gamma) = 1 \]
• Area of a Triangle in 3D: \(\text{Area} = \frac{1}{2}||\vec{AB} \times \vec{AC}||\)
• Area of Parallelogram from Determinant: \(\text{Area} = \sqrt{\det(A^T A)}\)

Linear Independence and Basis

• Linear Independence Condition: \(c_1\vec{v_1} + c_2\vec{v_2} + \dots + c_n\vec{v_n} = \vec{0}\) has only the trivial solution (\(c_1=c_2=\dots=c_n=0\)).
• Condition for Basis in \(\mathbb{R}^n\) using Determinant: The vectors \(\{\vec{v_1}, \dots, \vec{v_n}\}\) form a basis if \(det([\vec{v_1} \dots \vec{v_n}]) \neq 0\).
• Vector from Coordinates: \[ \vec{v} = c_1\vec{b}_1 + c_2\vec{b}_2 + \dots + c_n\vec{b}_n \]
• Change-of-Basis Transformation: \[ [\vec{v}]_{\mathcal{C}} = P_{\mathcal{C} \leftarrow \mathcal{B}} [\vec{v}]_{\mathcal{B}} \]
• Change-of-Basis Matrix Composition: \[ P_{\mathcal{C} \leftarrow \mathcal{B}} = \begin{bmatrix} [\vec{b}_1]_{\mathcal{C}} & [\vec{b}_2]_{\mathcal{C}} & \dots & [\vec{b}_n]_{\mathcal{C}} \end{bmatrix} \]
• Inverse Change-of-Basis: \[ P_{\mathcal{B} \leftarrow \mathcal{C}} = (P_{\mathcal{C} \leftarrow \mathcal{B}})^{-1} \]
• General Change-of-Basis via Standard Basis: \[ P_{\mathcal{B} \leftarrow \mathcal{C}} = (P_{\mathcal{S} \leftarrow \mathcal{B}})^{-1} P_{\mathcal{S} \leftarrow \mathcal{C}} \]

Matrix Properties

• Matrix Multiplication Element: \((AB)_{ij} = \sum_{k} a_{ik}b_{kj}\)
• Inverse of a 2x2 Matrix: For \(A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\), \[ A^{-1} = \frac{1}{ad-bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \]
• Matrix Inverse by Adjugate Method: \(A^{-1} = \frac{1}{\det(A)}\text{adj}(A)\)
• Determinant of a 2x2 Matrix: \(\det(A) = ad-bc\)
• Determinant of a Product: \(\det(AB) = \det(A)\det(B)\)
• Singular Matrix Condition: \(\det(A) = 0\)
• Orthogonal Matrix Condition: \(Q^T Q = I\) or \(Q^{-1} = Q^T\)
• Length Preservation by Orthogonal Matrix: \(||Qx|| = ||x||\)

Matrix Rank

• Rank-Nullity Theorem: \(\text{rank}(A) + \text{nullity}(A) = n\)
• Rank Sum Inequality: \(\text{rank}(A+B) \le \text{rank}(A) + \text{rank}(B)\)
• Rank Product Inequality: \(\text{rank}(AB) \le \min(\text{rank}(A), \text{rank}(B))\)
• Sylvester’s Rank Inequality: \(\text{rank}(A) + \text{rank}(B) - n \le \text{rank}(AB)\)

Systems of Linear Equations

• System of Equations: \(Ax = b\)
• Cramer’s Rule: \[ x_i = \frac{\det(A_i)}{\det(A)} \]
• Inverse Matrix Solution: \(x = A^{-1}b\)
• Condition for Unique Solution or Nonsingular Matrix: \(\det(A) \neq 0\)
• Condition for Nontrivial Solutions in a Homogeneous System: \(\det(A) = 0\)

Analytical Geometry in 2D

• General Form of a Line (2D): \(Ax + By + C = 0\) (where \((A, B)\) is the normal vector)
• Intercept Form of a Line (2D): \(\frac{x}{a} + \frac{y}{b} = 1\)
• Slope of a Line (2D): \(m = \frac{y_2 - y_1}{x_2 - x_1}\)
• Slope from Angle: \(m = \tan(\theta)\)
• Slope from General Form: \(m = -\frac{A}{B}\) (for \(Ax + By + C = 0\))
• Point-Slope Form (2D): \(y - y_1 = m(x - x_1)\)
• Perpendicular Slopes (2D): \(m_2 = -\frac{1}{m_1}\)
• Perpendicular Lines Condition (General Form): \(A_1A_2 + B_1B_2 = 0\) (for lines \(A_1x + B_1y + C_1 = 0\) and \(A_2x + B_2y + C_2 = 0\))
• Parallel Lines Condition (2D): \(\frac{A_1}{A_2} = \frac{B_1}{B_2} \neq \frac{C_1}{C_2}\)
• Coincident Lines Condition (2D): \(\frac{A_1}{A_2} = \frac{B_1}{B_2} = \frac{C_1}{C_2}\)
• Distance from a Point to a Line (2D): \(d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}\)

Analytical Geometry in 3D: Lines and Planes

• Parametric Equations of a Line: \[\begin{cases} x = x_0 + at \\ y = y_0 + bt \\ z = z_0 + ct \end{cases}\]
• Symmetric Equations of a Line: \[\frac{x - x_0}{a} = \frac{y - y_0}{b} = \frac{z - z_0}{c}\]
• Direction Vector from Intersecting Planes: \(\vec{v} = \vec{n}_1 \times \vec{n}_2\) (for line as intersection of two planes)
• Point-Normal Form of a Plane: \(a(x - x_0) + b(y - y_0) + c(z - z_0) = 0\)
• Standard Form of a Plane: \(ax + by + cz + D = 0\)
• Normal Vector from Cross Product: \(\vec{n} = \vec{u} \times \vec{v}\)
• Plane Through Three Points: Normal vector \(\vec{n} = \vec{M_1M_2} \times \vec{M_1M_3}\), then use point-normal form

Analytical Geometry in 3D: Distances and Angles

• Angle Between Two Lines: \[\cos \theta = \frac{|\vec{v_1} \cdot \vec{v_2}|}{||\vec{v_1}|| ||\vec{v_2}||}\] (using direction vectors)
• Angle Between Two Planes: \[\cos \theta = \frac{|\vec{n_1} \cdot \vec{n_2}|}{||\vec{n_1}|| ||\vec{n_2}||}\]
• Angle Between a Line and a Plane: \[ \sin(\theta) = \frac{|\vec{v} \cdot \vec{n}|}{||\vec{v}|| ||\vec{n}||} \]
• Distance from a Point to a Line (3D): \[D = \frac{||\vec{P_0 P} \times \vec{u}||}{||\vec{u}||}\]
• Distance from a Point to a Plane: \[\text{Distance} = \frac{|ax_1 + by_1 + cz_1 + D|}{\sqrt{a^2 + b^2 + c^2}}\]
• Distance Between Skew Lines: \[d = \frac{|\vec{P_1P_2} \cdot (\vec{v_1} \times \vec{v_2})|}{||\vec{v_1} \times \vec{v_2}||}\]

Conic Sections: General and Classification

• General Second-Degree Equation: \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\)
• Discriminant (Conic Classification): \(\Delta = B^2 - 4AC\)
  • \(\Delta < 0\): ellipse
  • \(\Delta = 0\): parabola
  • \(\Delta > 0\): hyperbola
• Matrix of Quadratic Form: \(A_{33} = \begin{bmatrix} A & B/2 \\ B/2 & C \end{bmatrix}\)
• Matrix of Quadratic Equation: \(A_q = \begin{bmatrix} A & B/2 & D/2 \\ B/2 & C & E/2 \\ D/2 & E/2 & F \end{bmatrix}\)

Conic Sections: Standard Forms

• Circle (Standard Form): \((x - h)^2 + (y - k)^2 = r^2\)
• Ellipse (Standard Form, Horizontal Major Axis): \(\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1\)
• Ellipse (Standard Form, Vertical Major Axis): \(\frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1\)
• Ellipse (Focus Relation): \(c^2 = a^2 - b^2\)
• Ellipse (Eccentricity): \(\varepsilon = \frac{c}{a} = \sqrt{1 - \frac{b^2}{a^2}}\)
• Hyperbola (Standard Form, Horizontal Transverse Axis): \(\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1\)
• Hyperbola (Standard Form, Vertical Transverse Axis): \(\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1\)
• Hyperbola (Focus Relation): \(c^2 = a^2 + b^2\)
• Hyperbola (Eccentricity): \(\varepsilon = \frac{c}{a} = \sqrt{1 + \frac{b^2}{a^2}}\)
• Hyperbola (Asymptotes, Horizontal): \(y - k = \pm \frac{b}{a}(x - h)\)
• Hyperbola (Asymptotes, Vertical): \(y - k = \pm \frac{a}{b}(x - h)\)
• Parabola (Standard Form, Vertical Axis): \((x - h)^2 = 4p(y - k)\)
• Parabola (Standard Form, Horizontal Axis): \((y - k)^2 = 4p(x - h)\)
• Parabola (Canonical Forms): \(y^2 = 4px\) or \(x^2 = 4py\)
• Directrix (Horizontal Ellipse): \(x = h \pm \frac{a}{e}\)
• Directrix (Horizontal Hyperbola): \(x = h \pm \frac{a^2}{c}\)
• Latus Rectum Length (Ellipse & Hyperbola): \(l = \frac{2b^2}{a}\)

Parametric Equations

• Circle (Parametric Form): \(x = h + r\cos\theta\), \(y = k + r\sin\theta\)
• Ellipse (Parametric Form): \(x = h + a\cos\theta\), \(y = k + b\sin\theta\)
• Pythagorean Identity: \(\cos^2\theta + \sin^2\theta = 1\)

Implicit Differentiation and Tangent Lines

• General Implicit Differentiation: For \(F(x, y) = 0\): \(\frac{dy}{dx} = -\frac{F_x}{F_y} = -\frac{\partial F/\partial x}{\partial F/\partial y}\)
• Perpendicularity Condition: \(m_1 \cdot m_2 = -1\)
• Tangent Length from External Point: For circle with center \(C\), radius \(r\), external point \(A\): \(\text{Tangent length} = \sqrt{(AC)^2 - r^2}\)

Coordinate Transformations and Rotations

• Coordinate Translation: \(x = x' + \alpha\), \(y = y' + \beta\)
• Rotation Angle (Cotangent): \(\cot 2\theta = \frac{A - C}{B}\)
• Rotation Angle (Cosine): \(\cos 2\theta = \frac{A - C}{\sqrt{(A - C)^2 + B^2}}\)
• Half-Angle Formulas:
  • \(\cos\theta = \pm\sqrt{\frac{1 + \cos 2\theta}{2}}\) (sign depends on \(\cot 2\theta\))
  • \(\sin\theta = +\sqrt{\frac{1 - \cos 2\theta}{2}}\) (always positive)
• Rotation Matrix: \(\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} \begin{bmatrix} x' \\ y' \end{bmatrix}\)
• Coordinate Rotation (Explicit): \(x = x' \cos\theta - y' \sin\theta\), \(y = x' \sin\theta + y' \cos\theta\)

Orthogonal Invariants

• Orthogonal Invariants (Ellipse/Hyperbola):
  • \(A + C = \tilde{A} + \tilde{C}\)
  • \(\det(A_{33}) = \tilde{A} \cdot \tilde{C}\)
  • \(\det(A_q) = \tilde{A} \cdot \tilde{C} \cdot \tilde{F}\)
• Center Coordinates: Solve \(\begin{cases} Ax_0 + \frac{B}{2}y_0 + \frac{D}{2} = 0 \\ \frac{B}{2}x_0 + Cy_0 + \frac{E}{2} = 0 \end{cases}\)
• Orthogonal Invariants (Parabola):
  • \(\tilde{C} = A + C\)
  • \(\tilde{D} = 2\sqrt{\frac{-\Delta}{A + C}}\) where \(\Delta = \det(A_q)\)

Quadric Surfaces in 3D

• Sphere (centered at origin): \(x^2 + y^2 + z^2 = r^2\)
• Sphere (general center): \((x - a)^2 + (y - b)^2 + (z - c)^2 = r^2\)
• Ellipsoid: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\)
• Elliptic Paraboloid: \(z = \frac{x^2}{a^2} + \frac{y^2}{b^2}\)
• Hyperbolic Paraboloid: \(z = \frac{x^2}{a^2} - \frac{y^2}{b^2}\)
• Hyperboloid of One Sheet: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1\)
• Hyperboloid of Two Sheets: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = -1\)
• Circular Cone: \(\frac{x^2}{a^2} + \frac{y^2}{a^2} = \frac{z^2}{c^2}\)

Linear Transformations

• Standard Matrix Construction: \(A = \begin{bmatrix} T(\mathbf{e}_1) & T(\mathbf{e}_2) & \cdots & T(\mathbf{e}_m) \end{bmatrix}\)
• Rotation Matrix (2D, counterclockwise by \(\theta\)): \(R = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}\)
• Scaling Matrix (2D): \(S = \begin{bmatrix} s_x & 0 \\ 0 & s_y \end{bmatrix}\)
• Reflection about x-axis: \(\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\)
• Reflection about y-axis: \(\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}\)
• Reflection about \(y = x\): \(\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\)
• Shear Matrix (horizontal): \(\begin{bmatrix} 1 & k \\ 0 & 1 \end{bmatrix}\)
• Projection onto xy-plane (3D): \(\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix}\)
• Projection onto x-axis (2D): \(\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}\)
• Composition of Transformations: \((S \circ T)(\mathbf{x}) = (BA)\mathbf{x}\) where \(T\) has matrix \(A\) and \(S\) has matrix \(B\)
• Rank-Nullity Theorem (Linear Transformations): \(\text{nullity}(T) + \text{rank}(T) = \dim(\text{Domain})\)

Polar Coordinates

• Polar to Cartesian Conversion: \(x = r \cos \theta\), \(y = r \sin \theta\)
• Cartesian to Polar Conversion: \(r = \sqrt{x^2 + y^2}\), \(\tan \theta = \frac{y}{x}\)
• Line Through Two Points in Polar: \(r[r_1 \sin(\theta - \theta_1) + r_2 \sin(\theta_2 - \theta)] = r_1 r_2 \sin(\theta_2 - \theta_1)\)
• Line Through Two Points in Polar (Alternative): \(\frac{1}{r} \sin(\theta_2 - \theta_1) = \frac{1}{r_2} \sin(\theta - \theta_1) - \frac{1}{r_1} \sin(\theta - \theta_2)\)
• Perpendicular Line in Polar: \(r[r_2 \cos(\theta - \theta_2) - r_1 \cos(\theta - \theta_1)] = r_0[r_2 \cos(\theta_0 - \theta_2) - r_1 \cos(\theta_0 - \theta_1)]\)

Spherical Coordinates

• Spherical to Cartesian Conversion: \(x = \rho \sin \varphi \cos \theta\), \(y = \rho \sin \varphi \sin \theta\), \(z = \rho \cos \varphi\)
• Cartesian to Spherical Conversion: \(\rho = \sqrt{x^2 + y^2 + z^2}\), \(\theta = \arctan\left(\frac{y}{x}\right)\), \(\varphi = \arccos\left(\frac{z}{\rho}\right)\)

Conic Sections: Eccentricity and Focus-Directrix

• Ellipse Eccentricity: \(e = \sqrt{1 - \frac{b^2}{a^2}} = \frac{c}{a}\) where \(c = \sqrt{a^2 - b^2}\) (for \(a > b\))
• Hyperbola Eccentricity: \(e = \sqrt{1 + \frac{b^2}{a^2}} = \frac{c}{a}\) where \(c = \sqrt{a^2 + b^2}\)
• Parabola Eccentricity: \(e = 1\) (always)
• Ellipse Foci (Horizontal Major Axis): \((\pm c, 0)\) where \(c = ae\)
• Ellipse Directrices (Horizontal Major Axis): \(x = \pm \frac{a}{e}\)
• Hyperbola Foci (Horizontal Transverse Axis): \((\pm c, 0)\) where \(c = ae\)
• Hyperbola Directrices (Horizontal Transverse Axis): \(x = \pm \frac{a}{e}\)
• Parabola Focus (Right-Opening): \((p, 0)\) for \(y^2 = 4px\)
• Parabola Directrix (Right-Opening): \(x = -p\) for \(y^2 = 4px\)

Conic Sections in Polar Form

• General Polar Form (Vertical Directrix): \(r = \frac{ed}{1 \pm e \cos \theta}\)
• General Polar Form (Horizontal Directrix): \(r = \frac{ed}{1 \pm e \sin \theta}\)
• Ellipse Polar Form: \(r = \frac{a(1 - e^2)}{1 + e \cos \theta}\) (focus at origin, right directrix)
• Hyperbola Polar Form: \(r = \frac{a(e^2 - 1)}{1 + e \cos \theta}\) (focus at origin, right directrix)
• Parabola Polar Form: \(r = \frac{2p}{1 + \cos \theta}\) (focus at origin, directrix \(x = -2p\))