3D Geometry Review Flashcards

Comprehensive vocabulary flashcards covering 3D geometry concepts including vector operations, line and plane equations, quadric surfaces, and coordinate transformations.

Magnitude of a Vector

The length of a vector $\mathbf{u}$, denoted as $||u||$, and calculated using the formula $\sqrt{u_1^2 + u_2^2 + u_3^2}$.

Unit Vector

A vector with a magnitude of 1, calculated using the formula $\hat{u} = \frac{\mathbf{u}}{||\mathbf{u}||}$.

Dot Product

The operation between two vectors $\mathbf{u}$ and $\mathbf{v}$ defined as $u_1v_1 + u_2v_2 + u_3v_3$.

Angle Between 2 Vectors

The value $\theta$ determined by the relationship $\cos(\theta) = \frac{\mathbf{u} \cdot \mathbf{v}}{||\mathbf{u}|| ||\mathbf{v}||}$.

Parallel Vectors

Two vectors $\mathbf{u}$ and $\mathbf{v}$ are parallel if and only if there exists some scalar $a \in \mathbb{R}$ such that $\mathbf{u} = a\mathbf{v}$.

Co-Linear Points

Three points P, Q, and R are co-linear if the component forms of vectors $\vec{PQ}$ and $\vec{PR}$ are scalars of one another.

Cross Product Area

The magnitude of the cross product $||\mathbf{u} \times \mathbf{v}||$ represents the area of the parallelogram with $\mathbf{u}$ and $\mathbf{v}$ as adjacent sides.

Volume of a Parallelepiped

The volume of a parallelepiped with adjacent edges $\mathbf{u}$, $\mathbf{v}$, and $\mathbf{w}$ is given by the triple scalar product $\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})$.

Parametric Equation of a Line

Equations representing a line in the form $x = x_1 + at$, $y = y_1 + bt$, and $z = z_1 + ct$, where $(x_1, y_1, z_1)$ is a point on the line and $\langle a, b, c \rangle$ is a parallel vector.

Symmetric Equation of a Line

A representation of a line in the form $\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}$, where $\langle a, b, c \rangle$ is the direction vector.

Standard Form of a Plane

The equation $a(x-x_1) + b(y-y_1) + c(z-z_1) = 0$, where $(x_1, y_1, z_1)$ is a point on the plane and $\mathbf{n} = \langle a, b, c \rangle$ is the normal vector.

Normal Vector

A vector $\mathbf{n} = \langle a, b, c \rangle$ that is perpendicular (normal) to a plane.

Sphere Equation

The equation $(x-h)^2 + (y-k)^2 + (z-j)^2 = r^2$, where $(h, k, j)$ is the center and $r$ is the radius.

Ellipsoid

A quadric surface with the general equation $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} + \frac{(z-j)^2}{c^2} = 1$, where $a$, $b$, and $c$ are not all equal.

Hyperboloid with Two Sheets

A quadric surface defined by equations where two variables are negative relative to the constant, such as $\frac{z^2}{c^2} - \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$.

Cone (Quadric Surface)

A quadric surface represented by equations such as $z^2 = x^2 + y^2$ or $\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 0$.

Cylindrical Coordinates

A 3D coordinate system $(r, \theta, z)$ where $r = \sqrt{x^2 + y^2}$ is the radial distance, $\theta = \tan^{-1}\left(\frac{y}{x}\right)$ is the angle, and $z = z$.