chapter 2 in person lecture 8/26 Vector Basics: Displacement, Components, and Addition

Vectors: quantity with both magnitude and direction; examples include force, velocity, and displacement. Energy is not a vector.
Representations: arrows (graphical), components (x and y), subscripts (e.g., $vx$, $vy$), or a vector with a hat (e.g., $\vec{v}$).
Displacement: the vector from the starting position to the final position: $\mathbf{d}{\text{net}} = \mathbf{d}{\text{final}} - \mathbf{d}_{\text{initial}}.$ Net displacement of multiple legs is the sum of individual displacements.

Vectors are drawn with tails (start) and heads (end); the arrow tip indicates the final position.
Components separate a vector into x- and y- parts: $\mathbf{v} = vx \hat{\mathbf{x}} + vy \hat{\mathbf{y}}.$ Then $vx = v \cos\theta, \quad vy = v \sin\theta.$ If only the magnitude is needed, use $|\mathbf{v}| = v.$
Gravity primarily affects the y-component in projectile motion; the x-component is constant (ignoring air resistance).

Tip-to-tail method: align the tail of the next vector to the tip of the previous; the net displacement goes from the starting tail to the final tip.
Parallelogram rule: the diagonal gives the resultant when adding two vectors.
General sum: $\mathbf{D}{\text{net}} = \mathbf{v}1 + \mathbf{v}_2 + \cdots.$

To add vectors, break into components: $\mathbf{v} = (vx, vy).$
Example: a velocity or displacement with components is easier to sum: $vx = v \cos\theta, \quad vy = v \sin\theta.$
In problems, solve for one unknown using a single-equation relation from the four basic equations (see below).

Core relationships (degrees are typical in this course):
- Pythagorean: $x^2 + y^2 = h^2$
- Sine: $\sin\theta = \frac{\text{opposite}}{h}$
- Cosine: $\cos\theta = \frac{\text{adjacent}}{h}$
- Tangent: $\tan\theta = \frac{\text{opposite}}{\text{adjacent}}$
When solving, look for an equation with one unknown and use the appropriate piece (opposite/adjacent) for the chosen angle; if a different angle (e.g., $\phi$) is used, the roles of opposite/adjacent may swap.
Degrees vs radians: most problems in this course use degrees; ensure calculator is in degree mode.

Example: a vector with magnitude $h=20$ cm and angle $\theta=30^\circ$:
- $vx = 20\cos 30^\circ \approx 17.32\text{ cm}, \quad vy = 20\sin 30^\circ = 10\text{ cm}.$
Example: displacement of 100 ft east:
- $\mathbf{d} = 100\text{ ft} \text{ east}$ or $dx = 100\text{ ft},\ dy = 0.$
Example: 100 ft at 60° north of east:
- $\mathbf{d} = 100\text{ ft at }60^\circ\text{ north of east}.$
Note: two vectors can have the same magnitude but different directions; the directions matter for the resultant.

Distinguish vectors (with arrows or components) from scalars.
When using a different angle (e.g., $\phi$), ensure you’re mapping opposite/adjacent correctly; the sine/cosine definitions depend on the chosen angle.
In many problems, you’ll convert to components first and then reassemble as needed.

Focus on: (1) expressing vectors in components, (2) using the tip-to-tail method, (3) applying Pythagoras and trigonometry to find components, (4) using equations with a single unknown to solve problems.
Later topics: projectile motion (gravity affects $y$-component, neglect air resistance), velocity-time and position-time graphs.

Upcoming assessments: chapters 1 and skill review; format includes calculation problems and conceptual questions; about 50 minutes in class.
Group work and presentations in class; study groups available via survey; accommodations available ahead of time.
A space club informational meeting and Excel zone representative upcoming for student resources.