Vectors and Coordinate Systems – Study Notes for Physics 1051

Vectors and Coordinate Systems

Introduction to Vectors

In physics, many quantities require both a magnitude and a direction for a complete description. These are called vectors. Understanding vectors is essential for describing motion and forces in more than one dimension.

• Scalar: A quantity fully described by a single number (magnitude) only. Examples: mass, volume, area, length.

• Vector: A quantity described by both magnitude (size) and direction. Examples: displacement, velocity, acceleration, force, electric field.

Example: The velocity of a train can be represented as a vector with a certain magnitude (e.g., 5 m/s) and a direction (e.g., east).

Displacement as a Vector

Displacement is a vector that represents the change in position of an object. It depends only on the initial and final positions, not on the path taken.

• The magnitude of displacement is the straight-line distance between the starting and ending points.

• The direction of displacement is from the initial to the final position.

• Displacement is the same regardless of the path taken between two points.

Example: If Sam walks 200 ft northeast from his house, his displacement is (200 ft, northeast), regardless of the actual path taken.

Addition of Vectors

Vectors can be added together to find a resultant vector. The most common method is the head-to-tail method.

• To add vectors geometrically, place them head-to-tail.

• The resultant vector (sum) is drawn from the tail of the first vector to the head of the last vector.

• If vectors are at right angles, use the Pythagorean theorem to find the magnitude:

$C = \sqrt{A^2 + B^2}$

• The direction can be found using trigonometry:

$\theta = \tan^{-1}\left(\frac{B}{A}\right)$

Example: Adding a 4 mi vector east and a 3 mi vector north gives a resultant of 5 mi at 37° north of east.

Zero Vector and Vector Subtraction

If a set of vectors forms a closed loop (head-to-tail arrangement returns to the starting point), their sum is the zero vector. Subtracting vectors is equivalent to adding a vector in the opposite direction.

• The zero vector has zero magnitude and no direction.

• Vector subtraction: $\vec{A} - \vec{B} = \vec{A} + (-\vec{B})$

Coordinate Systems

To describe vectors quantitatively, we use coordinate systems. The most common is the Cartesian coordinate system, with perpendicular x and y axes.

• The origin and orientation of axes can be chosen for convenience.

• Each vector can be broken into components along the axes.

Example: A vector 3 units east and 4 units north has components (3, 4).

Vector Components and Unit Vectors

Any vector in a plane can be expressed in terms of its components along the x and y axes:

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

• Unit vectors $\hat{i}$ and $\hat{j}$ have magnitude 1 and point in the positive x and y directions, respectively.

• The magnitude of a vector from its components:

$|\vec{A}| = \sqrt{A_x^2 + A_y^2}$

• The direction (angle θ with respect to x-axis):

$\theta = \tan^{-1}\left(\frac{A_y}{A_x}\right)$

Geometric vs. Component Representation

Vectors can be represented geometrically (arrows) or by their components. Component representation is especially useful for calculations and vector addition.

• To add vectors in component form, add their respective components:

$\vec{C} = \vec{A} + \vec{B} = (A_x + B_x)\hat{i} + (A_y + B_y)\hat{j}$

Changing Coordinate Axes

Sometimes, it is convenient to tilt the coordinate axes to align with a surface or direction of motion. The process of resolving vectors into components works the same way, but with respect to the new axes.

• Unit vectors and components are defined relative to the chosen axes.

• Always specify the orientation of your axes when solving problems.

Summary Table: Scalars vs. Vectors

Quantity Type	Definition	Examples
Scalar	Described by magnitude only	Mass, volume, area, length, temperature
Vector	Described by magnitude and direction	Displacement, velocity, acceleration, force

Key Formulas

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

• Direction (angle): $\theta = \tan^{-1}\left(\frac{A_y}{A_x}\right)$

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

• Law of cosines (for non-right triangles): $C^2 = A^2 + B^2 - 2AB\cos\theta$

Applications and Examples

• Describing motion in two or more dimensions (e.g., projectile motion, navigation).

• Analyzing forces acting at angles (e.g., tension in ropes, muscle forces in anatomy).

• Resolving vectors into components to simplify problem-solving.