What Is a Vector Quantity? Definition, Notation, and 10 Examples of Quantities With Magnitude and Direction
A vector quantity is a physical quantity that has both magnitude and direction. Examples include displacement, velocity, acceleration, and force. A vector is drawn as an arrow, and the arrow length represents the magnitude.
For a reliable reference, see OpenStax distinguishes vectors, which have magnitude and direction, from scalar quantities.
Vectors are essential when describing direction in motion; see Kinematics Equations: 5 Equations of Motion, Variables, and Solved Examples.
What Is a Vector Quantity?
A vector quantity is a physical quantity that requires both a magnitude and a direction for a complete description. OpenStax defines vector quantities as physical quantities that are specified completely by a number of units and a direction, and lists displacement, velocity, position, force, and torque as examples.
Examples of vector quantities include a displacement of 5 m east, a velocity of 20 m/s north, and a force of 10 N downward.
What Are the Magnitude and Direction of a Vector?
Magnitude is the size of the vector, and direction is the orientation of the vector. Magnitude is the length of a vector and is always a positive scalar quantity. Direction is often expressed as an angle of rotation from east, west, north, or south, or counterclockwise from due east.
How Is a Vector Quantity Represented?
A vector quantity is represented by an arrow. The arrow starts at the tail, which is the initial point, and ends at the head, which is the terminal point marked by an arrowhead. The arrow length shows the magnitude, and the arrowhead shows the direction.
OpenStax distinguishes vectors from scalars with a bold letter or an arrow over the letter. Table 1 lists 5 notations used for vectors.
| Notation | Symbol | Meaning |
|---|---|---|
| Boldface letter | v | Vector v |
| Arrow over a letter | v⃗ | Vector v |
| Magnitude | |v⃗| or v | Size of the vector, a non-negative number |
| Unit vector | v̂ | Vector of magnitude 1 along the direction of v |
| Component form | v_x î + v_y ĵ | Vector written with x and y components |
What Is the Difference Between Vector and Scalar Quantities?
A vector quantity has magnitude and direction, while a scalar quantity has magnitude only. Time, mass, distance, length, volume, temperature, and energy are examples of scalar quantities.
Table 2 compares scalar and vector quantities across 5 attributes.
| Attribute | Scalar quantity | Vector quantity |
|---|---|---|
| Description | Magnitude only | Magnitude and direction |
| Representation | A number with a unit | An arrow or a set of components |
| Paired example 1 | Distance of 2.0 km | Displacement of 2.0 km east |
| Paired example 2 | Speed of 20 m/s | Velocity of 20 m/s north |
| Paired example 3 | Mass of 5 kg | Weight of 49 N downward |
What Are 10 Examples of Vector Quantities?
Ten examples of vector quantities are displacement, position, velocity, acceleration, force, momentum, torque, electric field, magnetic field, and weight. Displacement, velocity, acceleration, and force are standard vector quantities in motion problems. Electric and magnetic fields are vector fields.
Table 3 lists the symbol, SI unit, and an example value for each vector quantity.
| Vector quantity | Symbol | SI unit | Example value |
|---|---|---|---|
| Displacement | d⃗ | Meter (m) | 5 m east |
| Position | r⃗ | Meter (m) | 3 m north of the origin |
| Velocity | v⃗ | Meter per second (m/s) | 20 m/s north |
| Acceleration | a⃗ | Meter per second squared (m/s²) | 9.8 m/s² downward |
| Force | F⃗ | Newton (N) | 10 N at 45° above the horizontal |
| Momentum | p⃗ | Kilogram meter per second (kg·m/s) | 50 kg·m/s east |
| Torque | τ⃗ | Newton meter (N·m) | 12 N·m along the +z axis |
| Electric field | E⃗ | Newton per coulomb (N/C) | 500 N/C eastward |
| Magnetic field | B⃗ | Tesla (T) | 0.5 T upward |
| Weight | W⃗ | Newton (N) | 49 N downward for a 5 kg object |
What Are the 5 Properties of Vectors?
The 5 properties of vectors are equality, negation, the zero vector, the unit vector, and scalar multiplication. Table 4 lists each property with a definition and an example.
| Property | Definition | Example |
|---|---|---|
| Equal vectors | Same magnitude and same direction | Two displacements of 5 m east |
| Negative vector | Same magnitude and opposite direction | −(5 m east) = 5 m west |
| Zero vector | Magnitude of 0 and no specific direction | Net displacement of a round trip |
| Unit vector | Magnitude of exactly 1 | î, ĵ, and k̂ along the x, y, and z axes |
| Scalar multiple | Magnitude scales by the factor, and a negative factor reverses the direction | 3 × (2 m/s east) = 6 m/s east |
Scalar multiplication stretches or compresses a vector by a number, and a negative scalar flips the direction.
How Do You Add Vector Quantities?
Add vectors by placing the tail of the second vector at the head of the first and drawing the resultant from the first tail to the last head. In the head-to-tail method, the resultant vector runs from the tail of the first vector to the head of the last vector. The parallelogram method is a second geometric method.
The component method adds vectors algebraically. The 5 steps are:
- Resolve each vector into x and y components.
- Add all x components to get R_x.
- Add all y components to get R_y.
- Calculate the magnitude: R = √(R_x² + R_y²).
- Calculate the direction: θ = tan⁻¹(R_y / R_x).
A displacement of 3 m east plus a displacement of 4 m north gives R = √(3² + 4²) = 5 m at θ = tan⁻¹(4 / 3) = 53.1° north of east. The net force on an object equals the vector sum of all forces on the object.
How Do You Find the Components of a Vector?
The components of a vector are v_x = v cos θ and v_y = v sin θ, where θ is the angle from the positive x axis. A force of 50 N at 30° has F_x = 50 × cos 30° = 43.3 N and F_y = 50 × sin 30° = 25 N.
How Do You Find the Magnitude of a Vector?
The magnitude of a vector is |v| = √(v_x² + v_y²) in 2 dimensions, and |v| = √(v_x² + v_y² + v_z²) in 3 dimensions. The vector 3î + 4ĵ has a magnitude of 5.
How Do You Subtract Vectors?
Subtract a vector by adding its negative: A − B = A + (−B). Reverse the direction of B, and then add the 2 vectors.
What Are the Dot Product and Cross Product?
The dot product multiplies 2 vectors to give a scalar, and the cross product multiplies 2 vectors to give a vector. Table 5 lists 4 vector operations with the result type, formula, and an example.
| Operation | Result type | Formula | Example |
|---|---|---|---|
| Vector addition | Vector | R = A + B | 3 m east + 4 m north = 5 m at 53.1° north of east |
| Scalar multiplication | Vector | c × A | 3 × (2 m/s east) = 6 m/s east |
| Dot product | Scalar | A · B = AB cos θ | Work W = F d cos θ = 10 N × 5 m × cos 60° = 25 J |
| Cross product | Vector, perpendicular to A and B | |A × B| = AB sin θ | Torque τ = r F sin 90° = 0.5 m × 20 N = 10 N·m |
Who Introduced the Vector Concept?
William Rowan Hamilton introduced the term vector around 1843. Hamilton coined the term vector around 1843 while presenting quaternions. Hamilton discovered quaternions on October 16, 1843, and Josiah Willard Gibbs and Oliver Heaviside later formulated the system of vector analysis that is now almost universally used.
Gibbs and Heaviside reduced the quaternion formalism to the vector algebra of dot products, cross products, and components in the 1880s. The modern system gained final acceptance around 1910.
What Are 4 Common Misconceptions About Vector Quantities?
Physics education research identifies 4 common misconceptions about vector quantities. Nguyen and Meltzer (2003) investigated the vector understanding of 2,031 physics students, covering vector addition, magnitude, and direction. The study used a seven-item quiz in all introductory general physics courses at Iowa State during the 2000/2001 academic year. A later study reports that many introductory physics students have difficulties in carrying out vector addition.
Table 6 pairs each misconception with the correct physics.
| Misconception | Correct physics |
|---|---|
| A vector quantity is the same as its magnitude | A vector includes a direction, and the magnitude is 1 part of the vector |
| Any quantity with a direction is a vector | A vector quantity also obeys the rules of vector addition, and electric current has a direction but is a scalar |
| Vectors add by adding their magnitudes | Vector addition accounts for direction, so 3 N and 4 N at 90° add to 5 N |
| The magnitude of a vector is negative | The magnitude is never negative, and a negative sign in a component shows direction |
Where Are Vector Quantities Applied?
Vector quantities are applied in 4 fields. The fields and their uses are:
- Navigation: plotting a course from displacement and velocity vectors
- Structural engineering: analyzing the force vectors on bridges and buildings
- Meteorology: describing wind with speed and direction
- Computer graphics: representing positions, directions, and lighting angles
People Also Ask About Vector Quantities
What is a vector quantity in simple terms?
A vector quantity is a measurement that has a size and a direction.
Is velocity a vector quantity?
Yes, velocity is a vector quantity because velocity has a magnitude and a direction.
Is speed a vector quantity?
No, speed is a scalar quantity because speed has magnitude only.
Is electric current a vector quantity?
No, electric current is a scalar quantity because currents do not follow the rules of vector addition, even though a current has a direction along a wire.
Are time and energy vector quantities?
No, time and energy are scalar quantities.
Can a vector have zero magnitude?
Yes, the zero vector has a magnitude of 0 and no specific direction.
What is the SI unit of a vector quantity?
A vector quantity uses the SI unit of the physical quantity it describes, such as m/s for velocity and N for force.
What Is a Vector Quantity? Summary of 5 Key Facts
The 5 key facts about vector quantities are:
- A vector quantity has both magnitude and direction.
- Vectors are drawn as arrows, with the arrow length showing the magnitude.
- Examples of vector quantities include displacement, velocity, acceleration, and force.
- Vectors add by the head-to-tail method or by components.
- Scalar quantities, such as distance, speed, and mass, have magnitude only.
Title tag: AP Macroeconomics Graphs: 7 Key Graphs, Labels, and Shifts
The University of the West Indies
My passion as a teacher and leader in education is to contribute to students’ enhanced learning by creation of a positive teaching/learning environment, where they are motivated to develop strong critical thinking, analytical and problem-solving skills.
As acting Vice-Principal and head of Physics, I have amassed five years of experience in leading meetings, implementing strategies to maintain discipline and school ethos, liaise with stakeholders in planning for future development of the institution; successfully negotiated the implementation of various projects for student development as well as supervising effective execution of such. For the past 19 years, I have incorporated discovery learning within the classroom through practical activities, problem solving and cooperative learning and have realized that students retention in Physics and Mathematics have increased significantly.
I am always cognizant that students have different learning styles, and as such teaching strategies during planning and delivery must take this crucial factor into consideration and be tailored to meet such differences.