I am reading the following notes on Clifford Algebra: http://www.av8n.com/physics/clifford-intro.htm#sec-addition And I have a confusion about bivector addition.
The geometric interpretation of a bivector is an oriented parallelogram formed by two vectors in given order. Here bivector addition is interpreted as joining two parallelograms edge-to-edge. But what if we have two bivector $a \wedge b$ and $x \wedge y$ such that $a$ is not $-y$? In the Fig 3. of the page above, $c$ has to cancel $w$ for it to make sense.
What is the general geometric interpretation of a bi-vector addition?
First, I would like to clear up a misconception: the geometric interpretation of a bivector is not a parallelogram at all, or any other shape; it is an oriented area in a plane. Bivectors are typically drawn as parallelograms because the area of that parallelogram is always equal to the magnitude of the bivector. However, the bivector as a mathematical object contains no information about shape, just as it contains no information about location.
Seen in this light, your question is akin to asking how vectors can add tip-to-tail if they’re always emanating from the origin. A vector viewed geometrically has a length and a direction, but not a location. Intuitively, that means that you can “slide” a vector anywhere in its space just so long as the start and end points stay the same distance apart in the same direction, as changing location doesn’t change the identity of the vector. However, moving one end of a vector while the other stays fixed isn’t allowed, as that would change the magnitude and/or direction of the vector, which are its essence.
Just so, a bivector viewed geometrically has an area, attitude, and orientation, but not a location or, crucially, a shape. Intuitively, not only can a bivector be freely “slid around” within space (so long as attitude and orientation are preserved), but it can be “reshaped” without changing the identity of the bivector in any way (so long as the area is preserved). Algebraically, bivectors in 3D have multiple factorizations. The geometric interpretation of this is that every bivector in 3D has multiple pairs of vectors in its plane that wedge together to create that bivector. This may be easier to visualize in 2D: by finding the area you may convince yourself that not only are $y \wedge -x$ and $(x + y) \wedge (y - x)$ identically equal to $x \wedge y$, but $(x + y) \wedge y$ is too.
Given this freedom to “reshape” any bivector as needed, the visual example in the notes you linked might be made more explicit by showing the two bivectors as parallelograms that don’t line up nicely, factoring them with respect to each other to find a common vector ($a$ and $w$ in that example) and only then proceeding to sum them visually. And indeed, in 3D, this process always works: any two bivectors in three-dimensional space can always be factored so that they have a common vector.
It should be noted that this process of intuitive geometry does not generalize to higher dimensions. Above three dimensions, not all bivectors can be expressed as the wedge of two vectors and not all pairs of bivectors have a vector in common. However, bivector addition is still valid in higher dimensions, and is computed the same way algebraically, it’s just that the sum is more difficult to visualize, partly because the geometric interpretation is less neat.
For more detail and an all-around better treatment, I recommend Making Sense of Adding Bivectors by James Smith.