Lecture 002

Vectors in $\mathbb{R}^2$

Initial Point: init point of vector Terminal Point: end of vector Notation: $\overrightarrow{u}$, $\overrightarrow{v}$, $\overrightarrow{w}$

• Example $\overrightarrow{PQ} = \langle{4-3, 2-1}\rangle = \langle{1, 1}\rangle$

• $\overrightarrow{O}$ is the zero vector

Vector Addition: $\overrightarrow{u} + \overrightarrow{v} = \langle{x_1 + x_2, y_1 + y_2}\rangle$ Scalar Multiplication: $k \times \overrightarrow{v} = <kx, ky>$ where $\overrightarrow{v} = (x, y)$ Magnitude: magnitude of $\|v\| = \sqrt[2]{v_x^2 + v_y^2}$

Vectors in $\mathbb{R}^3$

Z-axis: always pointing out from the paper (blue in $\langle{r, g, b}\rangle$) Standard Equation of a Sphere (surface only): a sphere centered at $(a, b, c)$ with radius $r$ is $(x-a)^2 + (y-b)^2 + (z-c)^2 = r^2$

unit vector: vector with magnitude 1

• $i = \langle{1, 0, 0}\rangle$

• $j = \langle{0, 1, 0}\rangle$

• $k = \langle{0, 0, 1}\rangle$

• $\langle{a, b, c}\rangle = a \overrightarrow{i} + b \overrightarrow{j} + c \overrightarrow{k}$

normalize vector: divide by its magnitude

vector in component form: $\langle{x, y, z}\rangle$ express using standard unit vector: i, j, k, 0