|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. The way you draw a vector is by drawing an arrow. Refer to Image 1. This arrow has a length and points in some direction.
When we 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. Although Image 1 is a perfect vector, it's unclear where it is positioned. Refer to Modified Image 1.
If we have an arrow we can now communicate with someone else about how much the vector goes in the x direction, the y direction and the z direction by using unit vectors. A vector is a combination of unit vectors that are placed along the the coordinate axes. These unit vector have length one. 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. Refer to Image 2 on the bottom left. We can express any 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: