Calculus • Coordinate Geometry • Vectors & 3D • Algebra
King's Rule: $$\int_a^b f(x)dx = \int_a^b f(a + b - x)dx$$
Linear DE: $$\frac{dy}{dx} + Py = Q \implies y e^{\int P dx} = \int Q e^{\int P dx} dx + C$$
Parabola $y^2 = 4ax$: Tangent $y = mx + a/m$, Directrix $x + a = 0$.
Ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$: Tangent $y = mx \pm \sqrt{a^2 m^2 + b^2}$.
Hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$: Tangent $y = mx \pm \sqrt{a^2 m^2 - b^2}$.
$$\det(\operatorname{adj} A) = |A|^{n-1}, \quad \operatorname{adj}(\operatorname{adj} A) = |A|^{n-2} A$$
$$\det(kA) = k^n |A|$$
Shortest distance between skew lines $\vec{r} = \vec{a}_1 + \lambda \vec{b}_1$ and $\vec{r} = \vec{a}_2 + \mu \vec{b}_2$:
$$d = \frac{|(\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2)|}{||\vec{b}_1 \times \vec{b}_2||}$$