Vectors and Motion in Two and Three Dimensions: Study Notes for Chapter 3 (Part 1)

Vectors and Scalars

Definitions and Examples

Physical quantities in physics can be classified as either vectors or scalars. Understanding the distinction is fundamental for analyzing motion and forces.

• Vector: Has both magnitude and direction. Examples: displacement, velocity, acceleration, force, momentum.

• Scalar: Has only magnitude. Examples: mass, time, temperature.

Vector Addition and Subtraction

Graphical Methods

Vectors can be added or subtracted using graphical techniques, which help visualize the resultant vector.

• One Dimension: Add or subtract magnitudes with appropriate signs.

• Tail-to-Head Method: Place the tail of the next vector at the head of the previous; the resultant is drawn from the tail of the first to the head of the last.

• Parallelogram Method: For two vectors, place them tail-to-tail and complete the parallelogram; the diagonal gives the resultant.

Example Table: Vector Addition Methods

Method	Description
Tail-to-Head	Sequentially connect vectors; resultant from start to end.
Parallelogram	Vectors from common origin; resultant is diagonal.

Adding Vectors by Components

Vectors in two or three dimensions are resolved into components along chosen axes (usually x, y, and z).

• Draw a coordinate system and mark axes.

• Resolve each vector into components using trigonometric functions:

Formulas: $V_x = V \cos \theta$ $V_y = V \sin \theta$ $V = \sqrt{V_x^2 + V_y^2}$ $\tan \theta = \frac{V_y}{V_x}$

• Add all x-components and y-components separately to get the resultant vector's components.

• Find the magnitude and direction using the above formulas.

Step-by-Step Recipe for Adding Vectors

1. Pick two axes (x, y).

2. Find the x and y components for each vector.

3. Add the x components; add the y components.

4. Find the magnitude of the sum using the Pythagorean theorem.

5. Find the angle of the sum with respect to the axes.

Unit Vectors

Definition and Usage

Unit vectors have magnitude 1 and indicate direction along coordinate axes.

• Common unit vectors: i (x-axis), j (y-axis), k (z-axis).

• Any vector can be written as: $\vec{V} = V_x \hat{i} + V_y \hat{j} + V_z \hat{k}$

Position and Displacement in 3D

Position Vector

The position vector locates a point in space relative to an origin.

• $\vec{r} = x \hat{i} + y \hat{j} + z \hat{k}$

Displacement Vector

Displacement is the change in position:

• $\Delta \vec{r} = \vec{r}_2 - \vec{r}_1$

• In components: $\Delta \vec{r} = (x_2 - x_1) \hat{i} + (y_2 - y_1) \hat{j} + (z_2 - z_1) \hat{k}$

Velocity in Two and Three Dimensions

Average Velocity

Defined as displacement divided by time interval:

• $\vec{v}_{ave} = \frac{\Delta \vec{r}}{\Delta t}$

• In components: $\vec{v}_{ave} = \frac{\Delta x}{\Delta t} \hat{i} + \frac{\Delta y}{\Delta t} \hat{j} + \frac{\Delta z}{\Delta t} \hat{k}$

Instantaneous Velocity

The velocity at a specific instant is the derivative of position with respect to time:

• $\vec{v} = \lim_{\Delta t \to 0} \frac{\Delta \vec{r}}{\Delta t} = \frac{d\vec{r}}{dt}$

• Always tangent to the trajectory (path).

• Magnitude: $v = \sqrt{v_x^2 + v_y^2 + v_z^2}$

Acceleration in Two and Three Dimensions

Average and Instantaneous Acceleration

• Average acceleration: $\vec{a}_{ave} = \frac{\vec{v}_2 - \vec{v}_1}{t_2 - t_1}$

• Instantaneous acceleration: $\vec{a} = \frac{d\vec{v}}{dt}$

• In components: $\vec{a} = a_x \hat{i} + a_y \hat{j} + a_z \hat{k}$

Tangential and Normal Acceleration

• Tangential acceleration: Change in speed along the path (tangent).

• Normal acceleration: Change in direction (perpendicular to path, toward center of curvature).

• Both components may exist simultaneously.

Projectile Motion

General Principles

Projectile motion is a classic example of two-dimensional motion under constant acceleration (gravity).

• Horizontal and vertical motions are independent.

• Horizontal motion: constant velocity ($a_x = 0$).

• Vertical motion: constant acceleration ($a_y = -g$).

Equations of Motion

Horizontal (x-axis)	Vertical (y-axis)
$a_x = 0$ $v_x(t) = v_{0x}$ $x(t) = x_0 + v_{0x} t$	$a_y = -g$ $v_y(t) = v_{0y} - g t$ $y(t) = y_0 + v_{0y} t - \frac{1}{2} g t^2$ $v_y^2 = v_{0y}^2 - 2g(y - y_0)$

Velocity Components

• $v_{0x} = v_0 \cos \theta_0$

• $v_{0y} = v_0 \sin \theta_0$

• $\theta_0$ is the launch angle with respect to the horizontal.

Equation of the Path

• Eliminate time to relate $y$ and $x$:

Formulas: $y = y_0 + v_{0y} \frac{x}{v_{0x}} - \frac{1}{2} g \left( \frac{x}{v_{0x}} \right)^2$ $y = (\tan \theta_0)x - \frac{g x^2}{2 (v_0 \cos \theta_0)^2}$

• This is the equation of a parabola.

Horizontal Range

• The range $R$ is the horizontal distance when the projectile returns to its original height.

Formula: $R = \frac{2 v_0^2 \sin \theta_0 \cos \theta_0}{g} = \frac{v_0^2 \sin 2\theta_0}{g}$

Examples and Applications

• Calculating displacement for a mail carrier using vector addition.

• Projectile motion problems: finding time of flight, range, and final velocity.

• Analyzing circular motion and average velocity.

Summary Table: Key Equations

Quantity	Equation
Vector Addition (components)	$V_R = \sqrt{V_{Rx}^2 + V_{Ry}^2}$
Angle of Resultant	$\tan \theta = \frac{V_{Ry}}{V_{Rx}}$
Position Vector (3D)	$\vec{r} = x \hat{i} + y \hat{j} + z \hat{k}$
Displacement	$\Delta \vec{r} = \vec{r}_2 - \vec{r}_1$
Average Velocity	$\vec{v}_{ave} = \frac{\Delta \vec{r}}{\Delta t}$
Instantaneous Velocity	$\vec{v} = \frac{d\vec{r}}{dt}$
Projectile Range	$R = \frac{v_0^2 \sin 2\theta_0}{g}$

Additional info:

• Relative motion and more advanced projectile problems are introduced but not fully covered in these notes.

• Students should be able to apply vector addition both graphically and analytically to solve real-world displacement and motion problems.