Calculus III: Vectors & Quadric Surfaces

Vocabulary flashcards covering quadric surfaces reference equations, essential vector formulas, and equations of lines and planes in 3D space.

Ellipsoid

A quadric surface given by the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$.

Elliptic Paraboloid

A quadric surface given by the equation $z = \frac{x^2}{a^2} + \frac{y^2}{b^2}$.

Hyperbolic Paraboloid (Saddle)

A quadric surface given by the equation $z = \frac{y^2}{b^2} - \frac{x^2}{a^2}$.

Cone

A quadric surface given by the equation $\frac{z^2}{c^2} = \frac{x^2}{a^2} + \frac{y^2}{b^2}$.

Hyperboloid of One Sheet

A quadric surface given by the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1$ ($1$ minus sign = $1$ sheet).

Hyperboloid of Two Sheets

A quadric surface given by the equation $-\frac{x^2}{a^2} - \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$ ($2$ minus signs = $2$ sheets).

Dot Product

A vector operation defined as $\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + u_3 v_3 = \|\mathbf{u}\| \|\mathbf{v}\| \cos(\theta)$, used for finding angles and checking orthogonality ($\mathbf{u} \cdot \mathbf{v} = 0$).

Cross Product

A vector operation computed as the determinant of a matrix with row 1 as $\mathbf{i}, \mathbf{j}, \mathbf{k}$, row 2 as $u_1, u_2, u_3$, and row 3 as $v_1, v_2, v_3$; used for finding perpendicular vectors and calculating the area of a parallelogram ($\|\mathbf{u} \times \mathbf{v}\|$).

Vector Projection of $\mathbf{u}$ onto $\mathbf{v}$

A vector defined by the formula $\text{proj}_{\mathbf{v}} \mathbf{u} = \left( \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{v}\|^2} \right) \mathbf{v}$.

Scalar Projection (Component of $\mathbf{u}$ along $\mathbf{v}$)

The magnitude component of $\mathbf{u}$ along $\mathbf{v}$, defined by the formula $\text{comp}_{\mathbf{v}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{v}\|}$.

Vector Equation of a Line

An equation representing a line in 3D space defined as $\mathbf{r}(t) = \mathbf{r}_0 + t \mathbf{d}$.

Parametric Equations of a Line

Equations representing a line in 3D space given by $x = x_0 + a t$, $y = y_0 + b t$, and $z = z_0 + c t$.

Equation of a Plane

An equation in 3D space given by $a(x - x_0) + b(y - y_0) + c(z - z_0) = 0$, where $\mathbf{n} = \langle a, b, c \rangle$ is normal to the plane.