The sum

The elements of degree one, called vectors, are added with the classic rule of the parallelogram, that is, the components are added algebraically.

For the elements of degree two, called bivectors, it is necessary to proceed with more caution.
A bivector is an oriented portion of the plane, therefore it is characterized by a magnitude (the measure of the area), a direction (given from the circulation) and the orientation in space. It sounds surprising, but the bivector has no shape !
In fact it can be thought of as a rectangle or a circle or any other shape useful for summing purpose.

Imagine that we need to add two bivectors A and B, which lie on two planes P and Q that meet along the line L.
We can reshape A and B so that they are generated by a common vector \pmb{c}, then:
A &=& \pmb{a} \wedge \pmb{c}
B &=& \pmb{b} \wedge \pmb{c}
It will then result
A + B = (\pmb{a} + \pmb{b}) \wedge \pmb {c}

Next to the concept of sum we can see, thanks to Marc ten Bosch page, the projection of a bivector to get its components. The outer product of the two vectors \pmb{a} and \pmb{b} generates a bivector in three-dimensional space (in blue). This can be projected into the fundamental planes by obtaining the three component bivectors (in orange, purple and yellow).

Try moving points a and b and see how the projections change. The components of the vectors can be fixed by writing them directly in the white boxes. It is also possible to rotate the viewpoint to better understand the geometric relationships between the elements.

\pmb{a} \wedge \pmb{b} = (a_x e_1 + a_y e_2 + a_z e_3) \wedge (b_x e_1 + b_y e_2 + b_z e_3) =

a_x b_x \, e_1\wedge e_1 + a_x b_y \, e_1\wedge e_2 + a_x b_z \, e_1\wedge e_3 \,+
a_y b_x \, e_2\wedge e_1 + a_y b_y \, e_2\wedge e_2 + a_y b_z \, e_2\wedge e_3 \,+
a_z b_x \, e_3\wedge e_1 + a_z b_y \, e_3\wedge e_2 + a_z b_z \, e_3\wedge e_3

Recalling the external products of the versors:

we will finally obtain that the blue bivector B has as components:

B_x_y = (a_x b_y - b_x a_y)
B_x_z = (a_x b_z - b_x a_z)
B_y_z = (a_y b_z - b_y a_z)

Exercise: verify that |a \wedge b|^2 = area_{xy}^2 + area_{yz}^2 + area_{zx}^2