Unit 1: Vector Quantities

This unit provides a fundamental understanding of physical quantities by distinguishing between those that only have magnitude and those requiring direction. It covers the essential techniques for representing vectors both algebraically and geometrically using scale drawings. Students will master the graphical methods for adding, subtracting, and resolving vectors into their horizontal and vertical components.

1.1 Scalars and Vectors

• A physical quantity is any number or set of numbers used for a quantitative description of a physical phenomenon.
• Scalars (or scalar quantities) are physical quantities that can be completely specified by a single number (magnitude) together with an appropriate unit of measurement.
• Examples of scalars include time, distance, speed, length, volume, temperature, mass, energy, and power.
• Vectors are quantities that require both magnitude and direction for their complete description.
• Examples of vectors include velocity, force, displacement, acceleration, momentum, impulse, weight, and electric field strength.

1.2 Vector Representations

• Algebraically, vectors are represented by a bold letter (e.g., A) or a letter with an arrow over it ($\vec{A}$).
• Geometrically, a vector is represented by an arrow or an arrow-tipped line segment.
• The length of the arrow represents the magnitude of the vector, and the arrow points in the specified direction.
• The initial point of the arrow is called the tail, and the final point is the head.

Process: Drawing Vectors Graphically

1. Decide upon an appropriate scale and write it down.
2. Determine the length of the arrow representing the vector by using that scale.
3. Draw the vector as an arrow, ensuring the arrow head is filled in.
4. Fill in the magnitude of the vector on the diagram.

Types of Vectors

• Zero vector (or Null vector): A vector with zero magnitude and no direction.
• Unit Vector: A vector that has a magnitude equal to one.
• Equal vectors: Vectors that have the same magnitude and the same direction.
• Negative of a vector: A vector that has the same magnitude but the opposite direction as a given vector.

1.3 Vector Addition and Subtraction

• Only vectors of the same kind can be added (e.g., two forces, but not a force and a velocity).
• The resultant of a number of vectors is the single vector whose effect is the same as the individual vectors acting together.
• Vector subtraction is the addition of the negative of a vector.
• To subtract $\vec{B}$ from $\vec{A}$ ($\vec{A} – \vec{B}$), flip the direction of vector $\vec{B}$ to make it $-\vec{B}$ and then add it to $\vec{A}$.+1

1.4 Graphical Method of Vector Addition

• Graphical methods use a ruler and protractor to determine the resultant of vectors drawn to scale.

Process: General Procedure for Graphical Addition

1. Decide on an appropriate scale and record it.
2. Pick a starting point and draw the first vector with the correct length and direction.
3. Draw subsequent vectors with appropriate lengths and directions.
4. Draw the resultant based on the specific rule being used (Triangle, Parallelogram, or Polygon).
5. Measure the length of the resultant and use the scale to convert it to magnitude.
6. Use a protractor to measure the vector’s direction.

Specific Graphical Laws

• Triangle method (head-to-tail method): Used for two vectors; the head of the first vector is joined to the tail of the second. The resultant is the third side of the triangle, drawn from the tail of the first to the head of the second.
• Parallelogram method: Used for two vectors; the tails of both vectors are joined to form adjacent sides of a parallelogram. The resultant is the diagonal drawn from the same origin point.
• Polygon method: Used for more than two vectors; each successive vector’s tail is placed at the head of the previous one. The resultant is the arrow connecting the tail of the first vector to the head of the last.

Special Cases for Resultants

1. Parallel vectors: When two vectors act in the same direction, the resultant magnitude is the sum of their individual magnitudes ($|R| = |A + B|$).
2. Anti-parallel vectors: When two vectors act in opposite directions, the resultant magnitude is the difference between their magnitudes ($|R| = |A – B|$) in the direction of the larger vector.
3. Perpendicular vectors: The magnitude is found using the Pythagorean theorem ($|R| = \sqrt{A^2 + B^2}$), and direction is found using $\theta = \tan^{-1}(B/A)$.

1.5 Vector Resolution

• A single vector can be broken down into a number of vectors that, when added, give the original vector.
• These individual vectors are called components, and the process is known as resolving into components.
• In a rectangular coordinate system, a vector is typically resolved into a horizontal component ($A_x$) and a vertical component ($A_y$).

Process: Graphical Method of Vector Resolution

1. Select a scale and draw the vector to scale in the correct direction.
2. Extend the x- and y-axes from the tail of the vector.
3. From the head of the vector, draw perpendicular projections to the x- and y-axes.
4. Draw the x-component ($A_x$) from the tail to the intersection on the x-axis.
5. Draw the y-component ($A_y$) from the tail to the intersection on the y-axis.
6. Measure the length of these components and apply the scale to find their magnitudes.

Trigonometric Method of Vector Resolution

• Horizontal Component: $A_x = A \cos \theta$.
• Vertical Component: $A_y = A \sin \theta$.
• Magnitude of Original Vector: $|A| [cite_start]= \sqrt{A_x^2 + A_y^2}$.
• Direction of Original Vector: $\theta = \tan^{-1}(A_y / A_x)$.

Key Terminology

• Physical quantity: A number used for quantitative description of a physical phenomenon.
• Scalars: Quantities specified only by magnitude.
• Vectors: Quantities specified by magnitude and direction.
• Scalar quantities: Another term for scalars.
• Tail: The initial point of a vector arrow.
• Head: The final (pointed) point of a vector arrow.
• Zero vector / Null vector: A vector with zero magnitude and no direction.
• Unit Vector: A vector with a magnitude of one.
• Equal vectors: Vectors with identical magnitude and direction.
• Negative of a vector: A vector with the same magnitude but opposite direction to the original.
• Resultant: The single vector representing the combined effect of several vectors.
• Triangle method / head-to-tail method: A graphical addition technique joining the head of one vector to the tail of the next.
• Parallelogram method: A graphical addition technique using two vectors as adjacent sides of a parallelogram.
• Polygon method: A graphical addition technique used for more than two vectors.
• Triangle law of vector addition: The specific rule stating the resultant is the third side of a triangle formed by two vectors.
• Parallelogram law of vector addition: The specific rule stating the resultant is the diagonal of a parallelogram formed by two vectors.
• Components: The individual vectors that sum to the original vector.
• Resolving into components: The process of breaking a single vector into its constituent parts.