|Gradients and Directional Derivatives|
|Change of Coordinate Systems|
|Math for Computer Graphics and Computer Vision|
Vectors are quantities that are specified by both a magnitude of length and a direction. Perhaps the simplest vector is a Euclidean vector, represented by an arrow in Euclidean space. Refer to Image 1. This arrow has a length and points in some direction.
The length of a vector, which is measurable, is called the magnitude. The magnitude is denoted by . The magnitude of a vector is a scalar quantity, a numerical value with no direction.
Image 1 is a vector. However, Image 1 is not very useful because it's unclear where it is positioned. To use vectors, we need to introduce the coordinate system. Assigning x and y coordinates to a vector allows us to be more precise when we talk about the location of a vector. Refer to Modified Image 1 on the left. In this updated image, we now know the x and y coordinates. Our vector is 6 units in the x direction and 8 units in the y direction.
Another way to define the position of a vector uses unit vectors. In the x-y Cartesian coordinate system, the vectors and are our unit vectors. In text books, unit vectors can be written in bold text or have a hat placed above the variables.
Unit vectors have a length of one. Any vector may be written in terms of our unit vectors and through scalar multiplication and addition (this is discussed in the graphical introduction below). For now, imagine the unit vector is a rubber substance that can stretch or shrink so we change change the magnitude of the unit vector such that it can express any vector in the coordinate system.
For example, can be written as the following:
We can write the vector in the Modified Image 1 as .
Unit vectors correspond to the x-y-z coordinate system. points along the x-axis points along the y-axis and points along the z-axis. This notation is used in physics, engineering and linear algebra. In order to give a vector's position using unit vectors, we write it as a combination of unit vectors that are placed along the the coordinate axes.
The notation is used to emphasize the vector nature of a vector while the coordinate notation is use to emphasize the point mature of a vector.
When we have a vector quantity we put an arrow on top of the labeling letter to remind us that it is a vector. It looks like so . One way of writing vectors is by components, like this: . For example, suppose we want to write a specific vector in components, and we know the vector goes 3 units in the x direction, 2 units in the y direction and 0 units in the z direction. Then we can simply write: . The components of the same vector can also be written as: . This vector has an x-component of 3, a y-component of 2 and a z-component of 0. This is shown in the image on the right.
With as any numerical values we can also write any vector in terms of standard unit vectors: . If we are given we know that vector A is 3 units in the x and y direction and 2 units in the z direction. This vector could equivalently be written as or . It is shown in the image below.
If a vector is still a bit abstract to you then think of a compass. The arrow has a certain length, this is our magnitude, and it points in any direction (north, south, east, and west).
Click here for a graphical introduction to vectors:
Click here for an algebraic introduction to vectors:
Click here for an interactive demonstration: