Vector Definition

When you think of a vector, many people probably call up Vector's definition (from Despicable Me). He says:It's a mathematical term. A quantity represented by an arrow with both direction and magnitude.

That's me—because I am committing crimes with both direction and magnitude! Reloaded skidrow. Oh yeah!Ok, but really. What's a vector? I like the following definition (and this is the definition I give to students in class).Vector: a quantity with more than one element (more than one piece of information).That isn't the best definition, but it is better than 'magnitude and direction.' Perhaps the best way to understand vectors is to look at some examples. Suppose I am in a room and I move around to different locations to measure the temperature. Temperature at a particular location has just one element (like 22° C).

Since temperature only has one piece of information, we call it a scalar. Other examples of scalars would be: mass, electric charge, power, electric potential difference.Now suppose I go around to different points in the room in order to determine the air flow. At each location, the air can be moving in three different directions (x,y,z). So, to really measure the air velocity at each location I would need 3 elements.

Vector - a variable quantity that can be resolved into components. Variable quantity, variable - a quantity that can assume any of a set of values. Cross product, vector product - a vector that is the product of two other vectors. Vector sum, resultant - a vector that is the sum of two or more other vectors.

We call this air velocity a vector (a 3 dimensional vector) because it has three pieces of information. Other examples of vectors would be: forces, electric fields, acceleration, displacement.Can you have vectors with more or less than 3 elements? In introductory physics courses, it is common to just look at vectors with only 2 dimensions (x and y) just to make things simpler. Also, you can have 4 or 5 or even more dimensions in vectors. The only problem with higher order vectors is that it's more difficult to visualize them in three dimensional space.

The Zero VectorHere's the real problem with the 'magnitude and direction' definition of a vector—the zero vector. Suppose you want to represent a displacement in 2 dimensions. If you start from the origin and move 3 meters in the x direction and -2 in the y direction, you could write that as. Can I find the magnitude of that vector? Yes, it's very easy to see that the vector has a magnitude of zero meters.

What about the direction? Well, if the displacement didn't actually go anywhere you can't really say which direction it was in. The best answer is to say that the direction is undefined. So, here is a case of a displacement with zero magnitude and an undefined direction.

Is it a vector? Am I just being picky about the definition of a vector?

The Zero Vector in Actual PhysicsThe zero vector is not zero. Just to be clear, I can write the following two equations. These are different quantities. You can not set a vector quantity equal to a scalar quantity. You just can't do it. However, it happens. This equation was in a very recent introductory physics text.

This is exactly how the equation was displayed in the text.The textbook was trying to show the idea of an object in equilibrium. In equilibrium, the total force on the object is the zero vector. However, this equation says that the total sum of the vector forces is just equal to zero (the scalar). Oh, maybe they are using 0 to represent the zero vector? That would be a plausible explanation if they didn't use arrows to represent other vectors.

No, the zero vector is still a vector. The best way to show this equation would be.