3D Geometry: Coordinates and Vectors

• Math, Geometry, 3D involves the study of shapes, sizes, relative positions of figures, and the properties of space in three dimensions.

Coordinates in 3D space

• A point in 3D space is located using three coordinates: x, y, and z.
• The 3D coordinate system is typically right-handed, meaning that if you curl the fingers of your right hand from the positive x-axis to the positive y-axis, your thumb points in the direction of the positive z-axis.
• The distance between two points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ is given by $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$.

Vectors in 3D space

• A vector in 3D space is represented as $\vec{v} = \langle a, b, c \rangle$, where a, b, and c are the components of the vector along the x, y, and z axes, respectively.
• Vector addition and scalar multiplication are performed component-wise.
• The dot product of two vectors $\vec{v} = \langle a_1, b_1, c_1 \rangle$ and $\vec{w} = \langle a_2, b_2, c_2 \rangle$ is given by $\vec{v} \cdot \vec{w} = a_1a_2 + b_1b_2 + c_1c_2$.
• The cross product of two vectors $\vec{v} = \langle a_1, b_1, c_1 \rangle$ and $\vec{w} = \langle a_2, b_2, c_2 \rangle$ is given by $\vec{v} \times \vec{w} = \langle b_1c_2 - c_1b_2, c_1a_2 - a_1c_2, a_1b_2 - b_1a_2 \rangle$. The cross product results in a vector that is perpendicular to both $\vec{v}$ and $\vec{w}$.

Lines in 3D space

• A line in 3D space can be represented parametrically as $\vec{r}(t) = \vec{r_0} + t\vec{v}$, where $\vec{r_0}$ is a position vector of a point on the line, $\vec{v}$ is the direction vector of the line, and $t$ is a parameter.
• Alternatively, a line can be defined by a point $(x_0, y_0, z_0)$ and a direction vector $\langle a, b, c \rangle$. The parametric equations of the line are then $x = x_0 + at$, $y = y_0 + bt$, $z = z_0 + ct$.
• Symmetric equations of a line are given by $\frac{x - x_0}{a} = \frac{y - y_0}{b} = \frac{z - z_0}{c}$.

Planes in 3D space

• A plane in 3D space can be defined by a point $(x_0, y_0, z_0)$ on the plane and a normal vector $\vec{n} = \langle a, b, c \rangle$ that is perpendicular to the plane.
• The equation of the plane is given by $a(x - x_0) + b(y - y_0) + c(z - z_0) = 0$, which can be written as $ax + by + cz + d = 0$, where $d = -(ax_0 + by_0 + cz_0)$.
• The angle between two planes is the angle between their normal vectors.

3D Shapes

• Common 3D shapes include spheres, cubes, cuboids, cones, cylinders, and pyramids.
• A sphere is the set of all points in 3D space that are equidistant from a center point. The equation of a sphere with center $(h, k, l)$ and radius $r$ is $(x-h)^2 + (y-k)^2 + (z-l)^2 = r^2$.
• A cylinder consists of all points that are a fixed distance from a central line, the axis of the cylinder.
• A cone is formed by a set of line segments, half-lines, or lines connecting a common point, the apex, to all the points on a base that is in a plane that does not contain the apex.

Transformations in 3D

• Transformations in 3D space include translation, rotation, scaling, and shearing.
• Translation involves shifting a 3D object along one or more of the x, y, and z axes.
• Rotation involves rotating a 3D object around one of the x, y, or z axes, or around an arbitrary axis. Rotation matrices are used to represent rotations.
• Scaling involves changing the size of a 3D object along one or more of the x, y, and z axes.
• Shearing involves distorting the shape of a 3D object by shifting points along one axis proportionally to their distance from another axis.

Surfaces in 3D

• Surfaces in 3D can be represented by equations involving x, y, and z.
• Examples include quadric surfaces such as ellipsoids, paraboloids, hyperboloids, and cones.
• Cylindrical coordinates $(r, \theta, z)$ and spherical coordinates $(\rho, \theta, \phi)$ can be used to describe surfaces in 3D. Conversion formulas exist between Cartesian, cylindrical, and spherical coordinates.
• Cylindrical coordinates: $x = r\cos\theta$, $y = r\sin\theta$, $z = z$.
• Spherical coordinates: $x = \rho\sin\phi\cos\theta$, $y = \rho\sin\phi\sin\theta$, $z = \rho\cos\phi$.

Calculus in 3D

• Functions of several variables: $f(x, y, z)$ maps a point in 3D space to a real number.
• Partial derivatives: The partial derivative of $f$ with respect to $x$ is denoted as $\frac{\partial f}{\partial x}$ and is found by treating $y$ and $z$ as constants.
• Gradient vector: $\nabla f = \langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \rangle$.
• Directional derivative: The directional derivative of $f$ in the direction of a unit vector $\vec{u}$ is given by $D_{\vec{u}}f = \nabla f \cdot \vec{u}$.
• Multiple integrals: Double and triple integrals are used to find the volume of 3D regions and to evaluate other quantities.