Vectors

Vectors are the most basic elements when it comes to 3D Math. In general mathematical terms, a vector is a one-dimensional array of numbers.

However, when it comes to animation, we mainly work with vectors of length 3 to represent the offsets along each axis of a coordinate system. Vectors are mainly used in many ways in 3D animation, from determining where a vertex should travel to how light bounces around a scene.

Many complex Maya tools and Deformers can be written with just some basic vector knowledge. In 3D Math, a vector consists of both a magnitude and a direction. The magnitude or length of a vector can be used to determine the distance between two points or how far an object has traveled.

Let’s take an example of two points forming a line at coordinates (1, 1, 0) and (3, 2, 0)

To calculate the distance between the two above points, we first subtract one point from the other in the order to find the vector between them

distance = (3, 2, 0) - (1, 1, 0) = (2, 1, 0)

Then we calculate the magnitude of that vector. To calculate the magnitude of a vector, we square each element, add them together, and take the square root. To calculate the magnitude of that above vector, we square each element, add them together and take the square root.

magnitude = sqrt(2 * 2 + 1 * 1 + 0 * 0)

We can scale a vector by multiplying it by a scalar value. This increases or decreases the magnitude. When we scale a vector by a negative value, the vector points in the opposite direction.

We can also add and subtract vectors to create new vectors. One thing about vectors is that they dont have any We can also add and subtract vectors to create new vectors. One thing about vectors is that they don’t have any positional data. Just for demonstration purposes, I am using Vector A and the end of Vector B, in reality, a vector is just a direction and a magnitude. If you want positional data, you should use a point. And if you want directional data, use a vector.

When a vector has a length of 1, we call it a Unit Vector. We can make any vector a unit vector by dividing each element by the vector’s magnitude. This is called normalizing the vector.

Normalized Vector = (3, 2, 0) / 3.61 = (0.83, .055, 0)

Dot Product

The dot product of two vectors A and B can be used to calculate the angle between the vectors or to find out if the vectors are pointing in the same direction.

The dot product is a single scalar value and is defined in two different ways.

First, the magnitude of Vector A times the magnitude of Vector B times Cosine of the Angle between the two vectors. Second, the sum of the products of the components of each vector, as seen in the equations below.

The dot product of two normalized vectors will always be between -1 and 1 if the dot product is greater than 0, the vectors are pointing in the same general direction. If the dot product is 0, the vectors are perpendicular, and if the dot product is less than 0, the vectors are pointing in opposite directions.

When we take the product of two vectors, A and B, and only one of the vectors is normalized, the resulting value is the length of the non-normalized vector projected onto the normalized vector. If we multiply the unit vector B by the stop product, the result is vector A projected onto vector B.

Cross Product

The cross-product of two vectors is a third vector that is perpendicular to both of them. Its magnitude is related to their magnitudes and the angle between them. Thus the cross product gives us a third perpendicular vector, we can create the axis of the Kornet system, which is two vectors.

To do this, we first calculate the cross-product between two vectors to get a third vector. Then we calculate another cross product between the new vector and one of the original vectors to make sure all the vectors are perpendicular to each other, these three vectors form the axis of the coordinate system.

Consider this animation, showing how the results of dot() and cross() change in relation to the changing angle:

This demonstrates two common applications of these methods. If the red vector is our object’s forward direction, and the green shows the direction towards another object:

• Dot product: Using the result, we can tell if the object is in front of (> 0) or behind (< 0) us.

• Cross product: Using the result, we can tell if the object is to the left (< 0) or right (> 0).

You can always find tonnes of free resources online if you need a bit more elaboration on each specific segment but this knowledge should help give you the jump start to write simple scripts, tools, or even deformers to fulfill some mathematical problems needed to solve complex deformations and movements in Animation and Visual Effects industry.

Thank you to many people who helped me along my way to understand 3D Math myself. Won’t be here if they did not help me learn and share their knowledge. Resources and images were either done by me or found online from Wikipedia and other related websites.

Until next time.

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