|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 and 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 is called the magnitude. The magnitude is denoted by . The magnitude of a vector is a scalar quantity, a numerical value.
Image 1 is a vector. However, Image 1 is not very useful because it's unclear where it the vector is positioned. We need to introduce the Caretesian coordinate system (the x-y graph) to properly give the direction of a vector. 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 locate a vector uses unit vectors. In the two-dimensional coordinate system, the vectors and are our unit vectors. It's clear that unit vectors have a length of one. In textbooks, unit vectors can be written in bold text or have a hat placed above the variables like so, .
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. With this property, 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 .
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. Unit vectors correspond to the x-y-z coordinate system this is in three-dimensions. points along the x-axis points along the y-axis and points along the z-axis. Unit vectors (sometimes called the standard basis vectors) are used in physics, engineering and linear algebra.
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: