8.2 Addition and Subtraction of Vectors
In vector mathematics, combining multiple displacements, velocities, or forces produces a single equivalent effect known as a resultant vector. Addition and subtraction of vectors depend on whether the vectors are parallel or non-parallel. Parallel vectors can be combined algebraically by considering their directions and signs, whereas non-parallel vectors are added or subtracted geometrically using the Triangle Law, the Parallelogram Law, or the Polygon Law. These operations provide essential mathematical tools to solve geometric configurations and practical problems involving navigation and motion.
Learning Objectives
- Define and identify a resultant vector as a single vector produced from the addition of two or more vectors.
- Perform addition and subtraction of parallel vectors and determine their magnitudes.
- Perform addition and subtraction of non-parallel vectors using the Triangle Law, Parallelogram Law, and Polygon Law.
- Apply vector subtraction by adding the negative vector: \(\underset{\sim}{a} - \underset{\sim}{b} = \underset{\sim}{a} + (-\underset{\sim}{b})\).
- Solve mathematical and real-life problems involving vector addition, subtraction, collinearity, and relative motion.
Geometric Laws of Vector Addition
Compare how resultant vectors are formed head-to-tail (Triangle and Polygon Laws) versus tail-to-tail (Parallelogram Law).
Triangle Law
Place the initial point of the second vector at the terminal point of the first vector. The resultant vector connects the starting point of the first vector directly to the terminal point of the second vector: \(\vec{AB} + \vec{BC} = \vec{AC}\).
Parallelogram Law
Two vectors \(\underset{\sim}{a}\) and \(\underset{\sim}{b}\) initiated from the same starting point form two adjacent sides of a parallelogram. The resultant vector is the directed diagonal from the common initial point to the opposite vertex: \(\vec{AC} = \vec{AB} + \vec{AD}\).
Polygon Law
For more than two vectors connected head-to-tail in succession, the resultant vector connects the starting point of the first vector directly to the terminal point of the last vector: \(\vec{AB} + \vec{BC} + \vec{CD} + \vec{DE} = \vec{AE}\).
Real-World Motion
When a boat is rowed due north at \(6\text{ km h}^{-1}\) and a river current flows due east at \(8\text{ km h}^{-1}\), the resultant velocity has magnitude \(\sqrt{6^2 + 8^2} = 10\text{ km h}^{-1}\) and direction \(053.13^\circ\).
Short Explanation
A resultant vector represents the net outcome of combining two or more vectors.
1. Resultant Vector
A resultant vector is the single vector that is produced from the addition of a few vectors. When an object travels through a sequence of displacements, the net displacement from the initial starting point to the final terminal point is the resultant vector.
2. Addition of Parallel Vectors
If two vectors act in the same direction, their magnitudes are added directly: \(4\underset{\sim}{a} + 5\underset{\sim}{a} = 9\underset{\sim}{a}\). The magnitude is \(|9\underset{\sim}{a}| = |4\underset{\sim}{a}| + |5\underset{\sim}{a}|\).
3. Subtraction of Parallel Vectors
Subtraction is defined as the addition of a negative vector: \(\underset{\sim}{a} - \underset{\sim}{b} = \underset{\sim}{a} + (-\underset{\sim}{b})\). When vectors are in opposite directions: \(7\underset{\sim}{b} - 4\underset{\sim}{b} = 7\underset{\sim}{b} + (-4\underset{\sim}{b}) = 3\underset{\sim}{b}\), so \(|3\underset{\sim}{b}| = |7\underset{\sim}{b}| - |4\underset{\sim}{b}|\).
4. Triangle Law
To add two non-parallel vectors, place the initial point of the second vector at the terminal point of the first vector (head-to-tail method): \(\vec{AB} + \vec{BC} = \vec{AC}\). Subtraction using the Triangle Law: \(\vec{AB} - \vec{CB} = \vec{AB} + (-\vec{CB}) = \vec{AB} + \vec{BC} = \vec{AC}\).
5. Parallelogram Law
Two vectors \(\underset{\sim}{a}\) and \(\underset{\sim}{b}\) initiated from the same starting point can be represented by two adjacent sides of a parallelogram, \(\vec{AB}\) and \(\vec{AD}\). The resultant vector \(\underset{\sim}{a} + \underset{\sim}{b}\) is the directed diagonal originating from the common initial point to the opposite vertex: \(\vec{AC} = \vec{AB} + \vec{AD}\).
6. Polygon Law
For more than two vectors connected head-to-tail in succession: \(\vec{AB} + \vec{BC} + \vec{CD} + \vec{DE} = \vec{AE}\). The resultant vector connects the starting point of the first vector directly to the terminal point of the last vector.
7. Commutative Property
Vector addition obeys the commutative law: \(\underset{\sim}{a} + \underset{\sim}{b} = \underset{\sim}{b} + \underset{\sim}{a}\).
8. Solving Problems Involving Vectors
For real-world motion, the resultant velocity is the vector sum of the object's velocity and the medium's velocity: \(\vec{v}_{\text{real}} = \underset{\sim}{v} + \underset{\sim}{a}\). When components are perpendicular, the magnitude is found using the Pythagorean theorem: \(|\underset{\sim}{v} + \underset{\sim}{a}| = \sqrt{v^2 + a^2}\). Direction is found using trigonometric ratios, such as \(\tan \theta = \dfrac{\text{opposite}}{\text{adjacent}}\), and stated as an angle or compass bearing in three-figure format, e.g., \(063.43^\circ\).
| Case | Method | Resultant Vector |
|---|---|---|
| Parallel vectors, same direction | Add magnitudes directly | \(4\underset{\sim}{a} + 5\underset{\sim}{a} = 9\underset{\sim}{a}\) |
| Parallel vectors, opposite direction | Add the negative vector | \(7\underset{\sim}{b} - 4\underset{\sim}{b} = 3\underset{\sim}{b}\) |
| Non-parallel vectors, head-to-tail | Triangle Law | \(\vec{AB} + \vec{BC} = \vec{AC}\) |
| Non-parallel vectors, tail-to-tail | Parallelogram Law | \(\vec{AC} = \vec{AB} + \vec{AD}\) |
| Three or more vectors, head-to-tail | Polygon Law | \(\vec{AB} + \vec{BC} + \vec{CD} + \vec{DE} = \vec{AE}\) |
Steps for Solving Vector Addition Problems
Identify whether the vectors are parallel or non-parallel.
For parallel vectors, combine scalar coefficients along the common direction.
For non-parallel vectors, apply the Triangle Law, Parallelogram Law, or Polygon Law.
For subtraction, add the negative vector: \(\underset{\sim}{a} - \underset{\sim}{b} = \underset{\sim}{a} + (-\underset{\sim}{b})\).
Vector Subtraction
\(\underset{\sim}{a} - \underset{\sim}{b} = \underset{\sim}{a} + (-\underset{\sim}{b})\)Subtraction is the addition of a negative vector.
Resultant Velocity Magnitude
\(|\underset{\sim}{v} + \underset{\sim}{a}| = \sqrt{v^2 + a^2}\)Use when the velocity components are perpendicular.
Try to Answer First
Answer in your mind, then press "Check Answer".
Given that \(\vec{PQ} = 3\underset{\sim}{a} + 2\underset{\sim}{b}\) and \(\vec{PS} = \underset{\sim}{b} - 2\underset{\sim}{a}\) are two adjacent sides of a parallelogram \(PQRS\) starting from point \(P\). Express the diagonal vector \(\vec{PR}\) in terms of \(\underset{\sim}{a}\) and \(\underset{\sim}{b}\).
In pentagon \(ABCDE\), \(\vec{AB} = \underset{\sim}{u}\), \(\vec{BC} = 2\underset{\sim}{v}\), \(\vec{CD} = 3\underset{\sim}{u}\), and \(\vec{DE} = -\underset{\sim}{v}\). Find the resultant vector \(\vec{AE}\) in terms of \(\underset{\sim}{u}\) and \(\underset{\sim}{v}\).
A boat is rowed due north at a velocity of \(6\text{ km h}^{-1}\). A river current flows due east at \(8\text{ km h}^{-1}\). Calculate the magnitude of the real velocity of the boat and state its direction as a bearing.
Common Mistakes
- Connecting vectors tail-to-tail instead of head-to-tail when using the Triangle Law.
- Forgetting to reverse the direction and sign when subtracting a vector, i.e., neglecting that \(-\vec{AB} = \vec{BA}\).
- Adding the scalar magnitudes directly for non-parallel vectors, e.g., writing \(|\underset{\sim}{a} + \underset{\sim}{b}| = |\underset{\sim}{a}| + |\underset{\sim}{b}|\) for perpendicular vectors instead of using the Pythagorean theorem.
- Misidentifying the diagonal in the Parallelogram Law — the resultant diagonal must start from the common initial point, not connect the two arrowheads.
- Writing compass bearings without the standard three-digit format, e.g., writing \(63.43^\circ\) instead of \(063.43^\circ\).
Concept Misunderstandings
Vector addition is not commutative.
Vector addition is commutative, meaning \(\underset{\sim}{a} + \underset{\sim}{b} = \underset{\sim}{b} + \underset{\sim}{a}\).
Vector subtraction \(\underset{\sim}{a} - \underset{\sim}{b}\) has its own unique geometrical rule unrelated to addition.
Vector subtraction is simply the addition of the negative vector, \(\underset{\sim}{a} + (-\underset{\sim}{b})\).
In the Triangle Law, the resultant vector points from the head of the second vector to the tail of the first vector.
The resultant vector always points from the starting point (tail of the first vector) to the ending point (head of the second vector).
Summary
- A resultant vector is the single vector formed by combining two or more vectors.
- For parallel vectors, addition and subtraction are performed by combining scalar coefficients along the common direction.
- For non-parallel vectors: Triangle Law: \(\vec{AB} + \vec{BC} = \vec{AC}\) (head-to-tail). Parallelogram Law: Two vectors from a common origin form adjacent sides; the diagonal from that origin is their resultant. Polygon Law: \(\vec{AB} + \vec{BC} + \vec{CD} + \dots + \vec{YZ} = \vec{AZ}\) for multi-step paths.
- Subtraction rule: \(\underset{\sim}{a} - \underset{\sim}{b} = \underset{\sim}{a} + (-\underset{\sim}{b})\).
- Real-world motion problems combine independent velocity vectors using vector addition and trigonometry.
Short Activity
Objective: Identify vector laws, subtraction rules and key terms in addition and subtraction of vectors.
A. Objective Quiz
1 Based on the Triangle Law of vector addition, which vector equation is correct for a triangle with vertices \(P\), \(Q\), and \(R\)?
2 Two vectors \(\underset{\sim}{a}\) and \(\underset{\sim}{b}\) share a common starting point and form two adjacent sides of a parallelogram. Which line represents the resultant vector \(\underset{\sim}{a} + \underset{\sim}{b}\)?
B. Fill in the Blanks
3 The single vector that is produced from the addition of a few vectors is called the ________ vector.
4 For any two vectors \(\underset{\sim}{a}\) and \(\underset{\sim}{b}\), vector subtraction is defined as adding the negative vector such that \(\underset{\sim}{a} - \underset{\sim}{b} = \underset{\sim}{a} +\) (________).
C. Matching / Drag and Drop
Match each vector law or operation to its correct mathematical expression or description. On a phone, tap a term first, then tap the matching box.
