# j-james/math

All my math notes, now in Markdown.

View the Project on GitHub j-james/math

# 3D Space and Vectors

## Learning Targets

You should be able to

### Part 1

• Plot points and graph planes
• Evaluate midpoint and distance of points
• Write equations of spheres
• Write vectors (component form / linear combination form)
• Find the unit vector
• Evaluate the dot product
• Write parametric / vector equations of lines

### Part 2

• Evaluate the angle between vectors
• Evaluate the cross product
• Write the equation of a plane
• Find the angle between planes
• Find the equation of the line of intersection of two planes
• Find the distance between a point and a plane

## Concepts / Definitions

### Plotting points in 3D

Sketch gridlines in the $xy$ plane and vertically to a point to show perspective. Make sure they are parallel. Right-hand rule axis.

### Graphing a plane

Plot intercepts, sketch the “triangle” which represents a plane in one octant.

The distance $d$ between the points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ is $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$$

The midpoint $M$ of line segment $PQ$ with endpoints $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ is $$(M_x M_y M_z) = (\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2})$$

The equation of a sphere with point $P(x, y, z)$ on the sphere and center $(h, k, l)$ and radius $r$ is $$(x-h)^2 + (y-k)^2 + (z-l)^2 = r^2$$

An equation of a plane can be written as $$Ax + By + Cz + D = 0$$ provided $A$, $B$, and $C$ are not all zero.

### Standard Unit vectors

$\vec{\imath} = \langle 1, 0, 0 \rangle,\ \vec{\jmath} = \langle 0, 1, 0 \rangle,\ \vec{k} = \langle 0, 0, 1 \rangle$

### Dot Product

$\vec{u} \bullet \vec{v} = u_x v_x + u_y v_y + u_z v_z$

The equation of a line through the point $P_0 (x_0, y_0, z_0)$ in the direction of $\vec{v} = \langle a, b, c \rangle$

Parametric form: $$x = x_0 + at, y = y_0 + bt, z = z_0 + ct$$

Vector form: $$\langle x, y, z \rangle = \langle x_0, y_0, z_0 \rangle + t \langle a, b, c \rangle$$

### Cross Product

If $\vec{u} = u_1 \vec{\imath} + u_2 \vec{\jmath} + u_3 \vec{k}$ and $v_1 \vec{\imath} + v_2 \vec{\jmath} + v_3 \vec{k}$ are two vectors in space, then the cross product is $$\vec{u} \times \vec{v} = (u_2 v_3 - u_3 v_2) \vec{\imath} + (u_3 v_1 - u_1 v_3) \vec{\jmath} + (u_1 v_2 - u_2 v_1) \vec{k}$$

$\vec{u} \times \vec{v}$ is a vector that is orthogonal to $\vec{u}$ and $\vec{v}$ $\lvert \vec{u} \times \vec{v} = \lvert\vec{u}\rvert \lvert\vec{v}\rvert \sin{\theta}$, where $\theta$ is the angle between $\vec{u}$ and $\vec{v}$.
$\lvert\vec{u} \times \vec{v} \rvert$ is the area of the parallelogram having non-zero vectors $\vec{u}$ and $\vec{v}$ as adjacent sides.

NOTE: Direction of vector is determined by the right-hand rule (order matters).

Torque (in physics) is a force, and is how much turning power you have. It is the cross product of the radius vector and force vector. Units are Newton-meter or lbs-ft, not same as work, which is energy, and Newton-meter is joule.
(Distance not in same direction, it’s perpendicular; multiply by unitless rotation (radians) to get work.)

The standard equation of a plane containing the point $P (x_1, y_1, z_1)$ and having nonzero normal (perpendicular) vector $\vec{n} = \langle a, b, c \rangle$ is $$a(x - x_1) + b(y - y_1) + c(z - z_1) = 0$$

The general form after being simplified is $$ax + by + cz + d = 0$$

The angle between two planes can be found by $$\cos{\theta} = \frac{n_1 \bullet n_2}{\lvert n_1\rvert \lvert n_2\rvert}$$ where $n_1$ and $n_2$ are the normal vectors of the two planes.

The equation of the line of intersection of the two planes, in parametric form, can be found by using the cross product of the two normal vectors of the planes and a point on the line. The cross product vector will be parallel to both planes, and will be in the direction of the line.

The distance between a plane and a point $Q$ (that is not in the plane) is $$d = \lvert proj_n\overrightarrow{PQ}\rvert$$ where $P$ is a point in the plane and $n$ is a vector normal to the plane.