Vector
A vector is a mathematical object that has a size, called the magnitude, and a direction.
For example, a vector would be used to show the distance and direction something moved in. If you ask for directions, and a person says "Walk one kilometer towards North", that's a vector. If he'd say "Walk one kilometer", without showing a direction, it would be a scalar.
We usually draw vectors as arrows. The length of the arrow is proportional to the vector's magnitude. The direction in which the arrow points is the vector's direction.
Examples of vectors
- John walks north 20 meters. The direction "north" together with the distance "20 meters" is a vector.
- An apple falls down at 10 meters per second. The direction "down" combined with the speed "10 meters per second" is a vector. This kind of vector is also called "Velocity".
Examples of things that are not vectors (Scalars)
- The distance between the two places is 10 kilometers. This distance is not a vector because it does not contain a direction.
- The number of fruit in a box is not a vector.
- A person pointing at a building is not a vector because there is only a direction. There is not a magnitude (the distance from the person's finger to the building, for example).
- The length of an object.
- A car drives at 100 kilometers per hour. That is not describing a vector, as there is only a number, but no direction.
More examples of vectors
- Displacement is a vector. Displacement is the distance that something moves in a certain direction. A measure of distance alone is a scalar.
- Force that includes direction is a vector.
- Velocity is a vector, because it is a speed in a certain direction.
- Acceleration is the rate of change of velocity. An object is accelerating if it is changing speed or changing direction.
How to add Vectors
Adding Vectors on paper using the Head to Tail method
The Head to Tail method of adding vectors is useful for doing an estimate on paper of the result of adding two vectors. To do it:
- Each vector is drawn as an arrow with an amount of length behind it, where each unit of length on the paper represents a certain magnitude of the vector.
- Draw the next vector, with the tail(end) of the second vector at the head(front) of the first vector.
- Repeat for all further vectors: Draw the tail of the next vector at head of the previous one.
- Draw a line from the tail of the first vector to the head of the last vector - that's the resultant(sum) of all the vectors.
It's called the "Head to Tail" method, because each head from the previous vector leads in to the tail of the next one.
See link for an example created with Java.[1]
Using Component Form
Using the component form to add two vectors literally means adding the components of the vectors to create a new vector. For example, let a and b be two two-dimensional vectors. These vectors can be written in terms of their components.
[math]\displaystyle{ \mathbf{a} = ( a_x, a_y ) }[/math]
[math]\displaystyle{ \mathbf{b} = ( b_x, b_y ) }[/math]
Suppose c is the sum of these two vectors, so that c = a + b. This means that [math]\displaystyle{ \mathbf{c} = ( a_x + b_x, a_y + b_y ) }[/math].
Here is an example of addition of two vectors, using their component forms.
[math]\displaystyle{ \mathbf{a} = ( 3, -1 ) }[/math]
[math]\displaystyle{ \mathbf{b} = ( 2, 2 ) }[/math]
[math]\displaystyle{ \mathbf{c} = \mathbf{a} + \mathbf{b} }[/math]
- [math]\displaystyle{ = ( a_x + b_x, a_y + b_y ) }[/math]
- [math]\displaystyle{ = ( 3 + 2, -1 + 2 ) }[/math]
- [math]\displaystyle{ = ( 5, 1 ) }[/math]
This method works for all vectors, not just two dimensional ones.