Vectors and Motion in One Dimension: Foundations of Physics with Calculus

Scalars and Vectors

Definitions and Key Differences

In physics, quantities are classified as either scalars or vectors. Scalars are described completely by a magnitude (number and unit), while vectors require both magnitude and direction for a full description.

• Scalar quantities: Examples include temperature, distance, area, volume, and speed.

• Vector quantities: Examples include velocity, displacement, and force.

Combining Scalars: Scalars are added or subtracted using ordinary arithmetic rules.

Combining Vectors: Vectors require both magnitude and direction to be considered in operations such as addition, subtraction, and multiplication by scalars.

Distance and Displacement

Understanding the Concepts

Distance is a scalar quantity representing the total length of the path traveled, regardless of direction. Displacement is a vector quantity representing the change in position from the initial to the final point, with both magnitude and direction.

• Distance: Total path length (scalar).

• Displacement: Straight-line vector from start to end position.

Magnitude of a Vector: The length of the vector, denoted as $|\vec{A}|$.

Vector Equality and Negation

Comparing and Reversing Vectors

Two vectors are equal if they have the same magnitude and direction, regardless of their initial points. The negative of a vector has the same magnitude but the opposite direction.

Vector Addition

Graphical Methods

Vectors can be added using the head-to-tail method or by constructing a parallelogram. The resultant vector extends from the tail of the first vector to the head of the last vector.

• Commutative property: $\vec{A} + \vec{B} = \vec{B} + \vec{A}$

• Parallelogram method: Place vectors tail-to-tail and complete the parallelogram; the diagonal is the sum.

Special Cases: Parallel and Antiparallel Vectors

When vectors are parallel, their magnitudes add. When they are antiparallel (opposite directions), their magnitudes subtract.

Multiplying Vectors by Scalars

Scaling Vectors

Multiplying a vector by a positive scalar changes its magnitude but not its direction. Multiplying by a negative scalar reverses the direction.

Vector Components

Breaking Vectors into Components

Any vector in a plane can be resolved into components along the x- and y-axes. The components are found using trigonometric functions:

• $A_x = A \cos \theta$

• $A_y = A \sin \theta$

Adding Vectors Using Components

To add vectors, sum their respective components:

• $R_x = A_x + B_x$

• $R_y = A_y + B_y$

Unit Vectors

Expressing Vectors in Component Form

Unit vectors $\hat{i}$, $\hat{j}$, and $\hat{k}$ point in the directions of the positive x-, y-, and z-axes, respectively, and have a magnitude of 1. Any vector can be written as a sum of its components multiplied by the corresponding unit vectors:

• $\vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k}$

Example: Vector Operations in Three Dimensions

Finding the Magnitude of a Vector Expression

Given $\vec{D} = (6.00 \hat{i} + 3.00 \hat{j} - 1.00 \hat{k})$ m and $\vec{E} = (4.00 \hat{i} - 5.00 \hat{j} + 8.00 \hat{k})$ m, find the magnitude of $2\vec{D} - \vec{E}$.

• First, compute $2\vec{D} = (12.00 \hat{i} + 6.00 \hat{j} - 2.00 \hat{k})$ m

• Then, $2\vec{D} - \vec{E} = (12.00 - 4.00)\hat{i} + (6.00 + 5.00)\hat{j} + (-2.00 - 8.00)\hat{k} = (8.00 \hat{i} + 11.00 \hat{j} - 10.00 \hat{k})$ m

• Magnitude: $|2\vec{D} - \vec{E}| = \sqrt{8.00^2 + 11.00^2 + (-10.00)^2} = \sqrt{64 + 121 + 100} = \sqrt{285} \approx 16.9$ m

Applications: Relative Velocity

Example: Airplane Velocities

When analyzing the motion of two airplanes, their velocities are vectors that can be added or subtracted to find relative velocity. For example, if plane A is moving at 500 km/h downward and plane B at 700 km/h at a 60° angle, their relative velocity can be found using vector addition.

Summary Table: Scalar vs. Vector Quantities

Quantity	Scalar or Vector	Example
Distance	Scalar	5 m
Displacement	Vector	5 m east
Speed	Scalar	60 km/h
Velocity	Vector	60 km/h north
Force	Vector	10 N right

Key Equations

• Magnitude of a vector: $|\vec{A}| = \sqrt{A_x^2 + A_y^2 + A_z^2}$

• Vector addition (components): $\vec{R} = (A_x + B_x)\hat{i} + (A_y + B_y)\hat{j} + (A_z + B_z)\hat{k}$

• Displacement: $\Delta \vec{x} = \vec{x}_f - \vec{x}_i$