|Gradients and Directional Derivatives|
|Change of Coordinate Systems|
|Math for Computer Graphics and Computer Vision|
Vectors are quantities that are specified by both a direction and magnitude of length. 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.
Although Image 1 is a vector, it 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 is by using unit vectors, which have length 1. 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. A unit vector is denoted by a lowercase letter with a hat above it, like so, . It is pronounced as a-hat. Our standard 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.
Referring to Image 2 on the right, we express a vector in terms of its components relying on the unit vectors. This helps us understand the direction. As for the magnitude, the length of a vector, we can just measure its length. The magnitude is denoted by . The magnitude of a vector is a scalar quantity, a numerical value with no direction.
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: . The components of a vector can also be written as: . This vector has an x-component of 3 and a y-component of 2, and 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 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: