What are Vectors and Scalars?

Types of Quantities in Physics Depend on Importance of Direction

Physics uses both scalar and vector quantities. Vectors include magnitude and direction; scalars include only magnitude.

Vectors

Vectors are defined as quantities that include both magnitude and direction. That means that completely specifying a vector quantity requires two numbers, one for the magnitude, or amount, and another for the direction.

Consider a wind weather report. One can report the wind speed by saying it is 20 miles per hour. Here the reported speed is not a vector quantity. It is a scalar. However just reporting the wind speed does not give all the information about the wind.

Someone wanting to go sailing will also want to know the wind direction. In this case, direction matters. A more complete wind velocity report might say that the wind is blowing 20 miles per hour towards the southwest. This wind velocity includes both the magnitude (20 miles per hour) and the direction (towards the southwest), so it is a vector quantity. (Parenthetical note: a wind blowing towards the southwest would be considered a northeast wind because winds are named for the direction of their origin. However velocity vectors point in the direction of travel.)

In physics velocity is a vector quantity, while the corresponding scalar quantity is speed. Velocity includes the direction; speed does not.

Scalars

Scalar quantities do not include direction. They include the magnitude only, and are represented by a single number.

In the example above, the wind speed is a scalar quantity. Even though the wind has a direction, only the magnitude is given. Some other quantities are scalars because the concept of direction makes no sense. For example, mass is a scalar rather than a vector quantity. The phrase “10 kilograms to the north” is nonsense. Mass does not have a direction. Therefore mass is a scalar quantity.

Representing Vectors Graphically

When drawing visualizations of problems, physicists represent vectors with arrows. The arrow points in the direction of the vector, and the length of the arrow is proportional to the magnitude of the vector quantity.

In the example of a wind that is 20 miles per hour towards the southwest, the wind velocity vector points towards the southwest. The scale of the drawing sets the length of the arrow.

Vectors are often given in terms of the magnitude and direction. The wind velocity vector is 20 miles per hour to the southwest. If the angle is measured from the east (a common convention in physics), the angle would be 225 degrees.

Vectors are also represented by their components. If the arrow representing the vector starts at the origin of Cartesian (x, y, & z) coordinates, then the x, y, & z coordinates of the point of the arrow represent the components of the vector. The two dimensional wind velocity vector above would have a x component of -14.1 miles per hour and a y component of -14.1 miles per hour.

Examples of Vector and Scalar Quantities

A few examples of vector quantities used in physics are: displacement, velocity, acceleration, force, momentum, angular velocity, and torque.

A few examples of scalar quantities are: mass, length, speed, energy, time, temperature, and charge.

Tensors

Both vectors and scalars are lower order examples of tensors. First year physics classes are usually told that moment of inertia is a scalar. In reality the moment of inertia depends on the rotational axis. To fully represent the moment of inertia about any possible axis requires a 3X3 matrix. The moment of inertia is therefore an example of a tensor quantity, which is more complex than either a vector or scalar quantity.

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