12.1 3D Coordinate System

Coordinate Axes: Perpendicular lines through the origin

• $x,y$ are horizontal, $z$ is vertical

Coordinate Plane: Planes determined by coordinate axes

• Octant: Like quadrant for 2D space, the coordinate planes divide up 3D space into 8 octants
• Coordinates: $(x,y,z) \in \R^3$
• Projection: Projection point obtained by perpendicular line from a point to a plane

Surfaces

Surface: Any shape created by an equation with $x,y,z$ in $\R^3$

• $x=k,y=k,z=k$ represents planes ($k$ is a constant)

Distance: $|P_1P_2|=\sqrt{(x_2-x_1)^2+(y_2 -y_1)^2+(z_2-z_1)^2}$

• Sphere: $(x-a)^2+(y-b)^2+(z-c)^2=r^2$ with origin $C(a,b,c)$

12.2 Vectors

Vector: Magnitude and direction

Displacement Vector: $\vec v = \vec{AB}$

• Zero Vector: $\vec 0$ has length 0 and no direction

Combining Vectors: $\vec{AC}=\vec{AB}+\vec{BC}$

• Scalar Multiple: $c\vec{v}$ have $c$ times the length and no change in direction

Vector Components: $\vec{v}=\langle a_1, a_2 \rangle$ from the origin

Position Vector: $\vec{OP}$ a vector representation from the origin to a point

• Given $A(x_a, y_a, z_a), B(x_b, y_b, z_b)$, $\vec{AB}=\langle x_b - x_a, y_b - y_a, z_b - z_a \rangle$

Vector Length: $||\vec{v}|| = \sqrt{v_1^2+v_2^2+v_3^2} = \sqrt{\vec{v} \cdot \vec{v}}$

• Unit Vector: $\vec u = \frac{\vec{v}}{|v|}$

Standard Basis Vectors: $\vec{i} = \langle1,0,0\rangle, \vec j = \langle 0, 1, 0 \rangle, \vec k = \langle 0, 0, 1 \rangle$

Dot Product: $\vec a \cdot \vec b = a_1b_1 + a_2b_2 + \dots + a_nb_n$

• $\vec a \cdot \vec b = |a| |b| \cos \theta$ ($\theta$ is the direction angle)