Vector Addition and Subtraction: Graphical Methods (OpenStax College Physics 2e, 3.2)

Vectors in Two Dimensions

A vector is a quantity that has both magnitude and direction. Examples include displacement, velocity, acceleration, and force.
In one dimension, direction is indicated by a plus or minus sign. In two dimensions (2-D), direction is specified relative to a reference frame (coordinate system) with an arrow whose length is proportional to the magnitude and pointing in the direction of the vector.
The text uses boldface symbols to denote vectors (e.g., \mathbf{F} for a force vector). The magnitude is denoted by the italic symbol (e.g., |\mathbf{F}|), and the direction is given by an angle \theta.
Example representation: a total displacement in 2-D can be described by its magnitude and direction, such as a displacement of 10.3 blocks at an angle \theta north of east.

Vector Notation and Graphical Representation

A vector is denoted by a boldface symbol, e.g., \mathbf{A}. Its magnitude is |\mathbf{A}| and its direction is given by an angle \theta.
A graphical vector is drawn as an arrow whose length is proportional to the magnitude and whose orientation matches the direction of the vector.
Example: A person walks 9 blocks east and 5 blocks north. The resultant displacement is 10.3 blocks at an angle \theta north of east.
Description of the resultant vector: to represent the total displacement graphically, draw an arrow for the vector with magnitude proportional to R and direction relative to the east-west axis.

Head-to-Tail Method (Graphical Vector Addition)

The head-to-tail method is a graphical way to add vectors.
Key definitions: tail is the starting point; head (tip) is the final point of a vector.
Steps (as in Figure 3.10 and follow-on figures):
1) Draw a vector representing the first displacement.
2) Draw a vector for the second displacement with its tail at the head of the first vector.
3) If there are more vectors, continue placing each new vector tail at the head of the previous one.
4) Draw a line from the tail of the first vector to the head of the last vector. This line is the resultant vector \mathbf{R}.
5) To obtain the magnitude of the resultant, measure its length with a ruler (often using the Pythagorean theorem when applicable).
6) To obtain the direction of the resultant, measure the angle it makes with the reference axis using a protractor (often using trigonometric relationships to determine the angle).
Accuracy: The accuracy of the graphical method is limited by drawing precision and measurement tools; it is valid for any number of vectors.
Important property: The vector sum is independent of the order in which vectors are added. Vector addition is commutative: \mathbf{A} + \mathbf{B} = \mathbf{B} + \mathbf{A} (and similarly for more vectors).

Example 3.1: Adding Vectors Graphically

Given three displacements (on a flat field):
- \mathbf{A} = 25.0 \mathrm{m} in a direction north of east.
- \mathbf{B} = 23.0 \mathrm{m} in a direction north of east.
- \mathbf{C} = 32.0 \mathrm{m} in a direction 68.0\° south of east.
Strategy: Represent each displacement as a vector, labeled A, B, and C, with lengths proportional to the distances and directions as specified relative to an east-west axis. Use the head-to-tail method to determine the resultant displacement \mathbf{R}.
Solution (summary):
1) Draw the three displacement vectors (Fig. 3.14).
2) Place the vectors head-to-tail retaining their magnitudes and directions (Fig. 3.15).
3) Draw the resultant vector \mathbf{R} (Fig. 3.16).
4) Measure the magnitude of \mathbf{R} with a ruler and the direction with a protractor.
5) In this example, the total displacement has a magnitude of \mathbf{R} = 50.8 \mathrm{m} and lies in a direction south of east (i.e., south of the eastward axis).
Result: \mathbf{R} = 50.8 \mathrm{m}, direction = south of east.
Discussion: The head-to-tail method works for any number of vectors, and the resultant is independent of the order of addition (commutative).

Vector Subtraction

Subtraction is a straightforward extension of addition: A - B is defined as A + (-B).
The negative of a vector \mathbf{B} is the vector -\mathbf{B}, which has the same magnitude as \mathbf{B} but points in the opposite direction.
Subtraction is equivalent to adding a negative vector, so the order of subtraction does not affect the final result in the same way as scalar subtraction.

Example 3.2: Subtracting Vectors Graphically

A = 27.5 \mathrm{m} in a direction north of east.
B = 30.0 \mathrm{m} in a direction north of east (or west of north). If the second leg is taken in the opposite direction, the second vector becomes -\mathbf{B}, which has the same magnitude 30.0 \mathrm{m} but points in the opposite direction (the problem describes the opposite as south of east).
Approach: To determine the end location after subtracting, draw vectors A and B, place them head-to-tail for A + (-B) to obtain the resultant R1. Then, to locate the dock, add A and B to obtain R2. Compare R1 and R2 to see how far the mistaken path lands from the dock.
Steps (summary):
1) Represent A and B as vectors.
2) Place vector B reversed (i.e., use -\mathbf{B}) to perform A + (-\mathbf{B}).
3) Draw the resultant R1 for the subtraction.
4) Represent the correct addition A + B to obtain R2 for the dock location.
5) Compare R1 and R2; the mistaken route generally ends up at a markedly different location from the dock.

Multiplication of Vectors by Scalars

Rules for scalar multiplication: If a vector \mathbf{A} is multiplied by a scalar c:
- The magnitude becomes |c| |\mathbf{A}|
- If c > 0, the direction does not change; if c < 0, the direction is reversed.
Example: If you multiply the first leg by 3, you get 3 × 25.0 \mathrm{m} = 82.5 \mathrm{m} in the same direction (north of east).
If the scalar is negative, the magnitude changes by |c| and the direction is reversed.
Division is the inverse of multiplication; dividing by a scalar c is equivalent to multiplying by 1/c. The same rules apply with c replaced by 1/c (assuming c ≠ 0).

Resolving a Vector into Components

Often we need to express a single vector as the sum of perpendicular components (usually along east-west and north-south axes).
For a vector of magnitude |\mathbf{A}| at an angle φ from the east toward the north (i.e., north of east):
- Ax = |\mathbf{A}| cos φ, \quad Ay = |\mathbf{A}| sin φ.
The components form a right triangle with the original vector as the hypotenuse.
Applications: Useful in projectile motion analysis and in dynamics for resolving forces and motions along perpendicular axes.
Analytical techniques for finding components are introduced in Vector Addition and Subtraction: Analytical Methods.

Quick Reference: Notation and Key Equations

Vector addition: \mathbf{R} = \mathbf{A} + \mathbf{B}.
Magnitude of the sum (general case): |
\mathbf{R}| = \sqrt{|\mathbf{A}|^2 + |\mathbf{B}|^2 + 2|\mathbf{A}|\,|\mathbf{B}|\cos \theta}, where \theta is the angle between \mathbf{A} and \mathbf{B}.
Special case (perpendicular vectors): |
\mathbf{R}| = \sqrt{|\mathbf{A}|^2 + |\mathbf{B}|^2}.
Subtraction: \mathbf{A} - \mathbf{B} = \mathbf{A} + (-\mathbf{B}).
Negative vector: -\mathbf{A} has the same magnitude as \mathbf{A} but points in the opposite direction.
Scalar multiplication: For a scalar c, |c\mathbf{A}| = |c| |\mathbf{A}|, and the direction is the same as \mathbf{A} if c > 0, or opposite if c < 0.
Component form: If the vector A has magnitude |A| and direction φ from the east toward the north, then Ax = |A| cos φ and Ay = |A| sin φ.
Directions such as north of east imply φ is measured from the east axis toward the north.

Connections and Practical Relevance

Resolving vectors into components is foundational for analyzing projectile motion and forces in dynamics.
The graphical head-to-tail method provides intuition and a stepping stone to the analytical methods for vector operations.
The commutative property of vector addition (A + B = B + A) underpins many problem-solving strategies and confirms that the order of vector addition does not affect the resultant.

Optional Resources

PHET Explorations: Maze Game for position, velocity, and acceleration concepts.
OpenStax College Physics 2e – Section 3.2: Vector Addition and Subtraction: Graphical Methods.