Amazing Things You Can Do in Geometric Algebra Explained

Transcript

vectors and scalers vectors are one of the main subjects of study in geometric algebra just like in linear algebra a vector can be thought of as a geometric object an arrow with some length and some direction the start of a vector is called its tail and the end of a vector is known as its head a vector's length is called its magnitude the direction and magnitude are the only properties defining the vector if two vectors have the same direction and the same magnitude they are the same vector regardless of where you draw them a vector can exist in any number of Dimensions including more than three dimensions even if that's hard to visualize the zero Vector is a special Vector with no direction and a magnitude of zero a vector can be represented by a list of numbers for example in two Dimensions assume that a vector's tail is placed at the origin the list of numbers for that Vector is the coordinates of its head for instance the vector represented by 32 points 3 units to the right and two units upward vectors can model real world phenomena for instance displacement is a vector quantity Where the vector in question points from an object's starting position to its current position another vector quantity is velocity the direction tells you which way an object is going and the magnitude tells you how fast meanwhile the word scaler is just a fancy way to say number this is in reference to their primary role in linear algebra as covered in the following section scalar Vector multiplication and vector addition in linear algebra scalars can be multiplied by vectors in order to scale them for instance a vector can be multiplied by the scaler 2 this has the effect of stretching the vector to double its original length keeping its direction the same scalers can also be used to scale down Vector like 1 over two squishing the vector to half its original length if a vector is multiplied by a negative scaler it reverses Direction multiplying a vector by -3 flips it around and triples its length in order to multiply a scalar and a vector simply multiply each of the vector's components by that scaler for instance using the vector 32 scaling it by two yields the vector 64 if you add two vectors you get another Vector in order to add them place the second one's tail on the first one's head then draw the vector from the first one's tail to the second one's head algebraic L adding two vectors simply involves adding their components for instance we could try adding the vectors 2 1 and 34 geometrically we see that this lands us at the point 55 algebraically 2 + 3 is 5 and 1 + 4 is 5 so we land at 55 once again if you prefer you can imagine adding two vectors as traveling along the first one then along the second the total displacement resulting from the journey is the sum using the previous two operations we can obtain any Vector within a space by simply starting with a given finite set of vectors for any number of Dimensions n we require exactly n vectors such a set of vectors is called a basis of the space and the space itself is called the span of the basis for Simplicity let's start with two-dimensional space in geometric algebra our standard basis vectors are unit vectors having a length of one in 2D these vectors are e 1 pointing right and E2 pointing up now let's try to represent the vector 2 1.5 we need to go left by two and up by 1.5 so the result is 2 E1 + 1.5 E2 in general a vector v with components A1 and A2 can be written as V equals A1 * E1 + A2 * E2 dot product the DOT product of two vectors is the sum of the products of their corresponding components we can illustrate with an example let's say that two vectors are defined as follows V = 2 E1 + 5 E2 and W = 2 E1 + E2 to find the dotproduct V * W find the first component of each Vector those being 2 and -2 here and multiply them giving -4 then take the next components the first vector has a five and the second has an implicit component of 1 multiplying to 5 finally -4 + 5 = 1 so the dotproduct of these two vectors is one the dotproduct has a certain geometric interpretation take the projection of V onto W which is another Vector that tells you how much of V is pointing in the same direction as W then take the magnitude of the projection and the magnitude of w and multiply them the result is the dotproduct of V and W this can also be done in reverse order with W being projected onto V instead to obtain the same result hence the dot product satisfies the commutative property another formula can be used to obtain the dotproduct of two vectors shown here the absolute value symbols denote the magnitude of each vector and Theta denotes the angle between the two vectors wedge product previously all discussions have been about Concepts applicable in linear algebra the wedge product is the first one that is exclusive to geometric algebra the wedge product of two vectors produces an entirely new form of object the B Vector the B Vector can be thought of as an oriented plane segment just like the vector can be thought of as an oriented line segment the by Vector can be oriented clockwise or counterclockwise and it also has an attitude defined by the particular plane that it is parallel to the only aspects of the B Vector that matter are the orientation attitude and magnitude that being the bi vector's area in this case if two B vectors have the same of all of these then they are one and the same in order to take the wedge product of two vectors V and W put their tails in the same position then draw out a parallelogram with the vectors as two of its sides with the wedge product order matters the vector that comes first determines the orientation of the V Vector traveling either clockwise or counterclockwise along the parallelogram this oriented parallelogram is a bi Vector though the specific shape actually doesn't matter only the orientation and magnitude the wedge product is closely related to an operation from linear algebra the cross product the cross product and wedge product are primarily useful in the same places however the wedge product is much more versatile not being restricted to work only in exactly three dimensions like the cross product one example of this is torque imagine using an usually long wrench 100 cm in length to screw a bolt counterclockwise out of the floor the more force your hand applies the faster the bolt will come out this is also affected by how far away you put your hand from the bolt where the bolt is called the axis of rotation the farther the better torque is a quantity that is affected by both the force Vector F and the position Vector pointing from the axis of rotation to the point where force is applied R this quantity is tradition Ally expressed as a cross product R * F producing a vector that points along the axis of rotation however using geometric algebra it can be expressed as a wedge product instead R wedge F this produces a b Vector that satisfies all of the properties we would want torque to have it increases when more force is applied it increases when the point of force application gets farther from the axis of rotation and it specifies an orientation however in contrast to the cross product there is no need to invoke a third dimension to make this work furthermore it is well behaved under typical Transformations such as Reflections this is one of many cases where geometric algebra proves useful as a way of describing physics geometric product one of the main aspects of geometric algebra separating it from linear algebra is the ability to multiply two vectors together the geometric product of two vectors V and W denoted by VW is equal to the sum of their dot product and wedge product of course one thing May jump out at you immediately the dot product produces a scaler and the wedge product produces a b Vector how can you add the two this does not really seem to be algebraically possible but we can simply accept that we are able to do this similarly to how we can accept that a real number plus an imaginary number is a complex number returning to the basis vectors in two Dimensions E1 and E2 two we can take the geometric product of the two vectors the dotproduct of two perpendicular vectors is just zero so this simplifies thus the geometric product of these basis vectors is a bi Vector called the unit bi Vector in higher dimensions a tri Vector four Vector five Vector Etc are possible any combination of these is called a multiv vector these are the main objects of study in geometric algebra